SAMPLE. A Publication of Complete Curriculum. Gibraltar, MI Complete Curriculum

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2 TM A Publication of Complete Curriculum Gibraltar, MI 2015 Complete Curriculum All rights reserved; No part of this publication may be reproduced or transmitted in any form or by any means without prior permission from the Publisher or Authorized Agent. Yearly memberships to access over 8,000 lessons, available at Published in electronic format in the U.S.A.

3 Acknowledgments Complete Curriculum s K-12 curriculum has been team-developed by a consortium of teachers, administrators, educational and subject matter specialists, graphic artists and editors. In a collaborative environment, each professional participant contributed to ensuring the quality, integrity and effectiveness of each Compete Curriculum resource was commensurate with the required educational benchmarks and contemporary standards Complete Curriculum had set forth at the onset of this publishing program.

4 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 1 Introduction to Place Value Objective: The student will be able to understand the place value of a whole number. Lesson 2 Understanding Millions and Billions Objective: The student will be able to understand the place value of digits in the millions and billions. Lesson 3 Important Whole Numbers Objective: The student will be able to estimate values using specific important numbers, called benchmark numbers. Lesson 4 Comparing and Ordering Whole Numbers Whole Numbers Assessment Objective: The student will be able to put whole numbers in numerical order from least to greatest by comparing the value of each whole number. Lesson 5 Introduction to Decimal Place Value Objective: The student will be able to understand and use the place value of decimals. Lesson 6 Finding Decimal Equivalents Objective: The student will be able to understand and write equivalent decimals. Lesson 7 Ordering and Comparing Decimals Decimals Assessment Objective: The student will be able to order and compare decimals. Lesson 8 Rounding Whole Numbers Objective: The student will be able to round whole numbers to a particular place value using his understanding of place value. Lesson 9 Rounding Decimals Objective: The student will be able to round decimals to a specific place value. Lesson 10 Estimating Objective: The student will be able to estimate the sum or difference in a problem using whole numbers or decimals. Lesson 11 Addition and Subtraction of Whole Numbers Objective: The student will be able to add and subtract whole numbers.

5 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 12 Addition and Subtraction with Decimals Addition and Subtraction with Decimals Assessment Objective: The student will be able to add and subtract with decimals Lesson 13 Introduction to Expressions and Variables Objective: The student will be able to write and solve algebraic expressions using variables. Lesson 14 Learning How to Write Equations Objective: The student will be able to write basic algebraic equations. Lesson 15 Learning How to Solve Equations Objective: The student will be able to solve basic algebraic equations using mental math. Lesson 16 Understanding Inequalities Objective: The student will be able to understand and use basic inequalities. Lesson 17 The Algebraic Properties Introduction to Algebra Assessment Objective: The student will understand the algebraic properties, and be able to distinguish what property is being used in an expression. Lesson 18 Finding Patterns Objective: The student will be able to find patterns using multiples of 10. Lesson 19 Multiplication of 1-Digit Numbers Objective: The student will be able to multiply numbers by a 1-digit number. Lesson 20 Multiplication by a 2-Digit Number Multiplication Assessment Objective: The student will be able to complete multiplication problems in which all factors have two or more digits. Lesson 21 Patterns with Decimal Multiplication Objective: The student will be able to use various patterns to multiply decimals. Lesson 22 Multiplying with Decimals Objective: The student will be able to multiply decimals.

6 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 23 More Multiplication with Decimals Objective: The student will be able to solve word problems involving multiplication with decimals. Lesson 24 Understanding Zeros Multiplying with Decimals Assessment Objective: The student will be able to multiply decimals that have zeros in the final answer. Lesson 25 Estimating Quotients Objective: The student will be able to use compatible numbers to determine an estimate for a division problem. Lesson 26 Dividing by 1-Digit Divisors Objective: The student will be able to find a quotient of a division problem with a 1-digit divisor. Lesson 27 Using Zeros in Division Objective: The student will be able to use zeros in division problems correctly. Lesson 28 Dividing by 2-Digit Divisors Objective: The student will be able to complete a division problem involving a 2-digit divisor. Lesson 29 More Practice with Division Division Assessment Objective: The student will be able to practice division by a 1 and 2 digit divisor. Lesson 30 Using Patterns When Dividing Decimals Objective: The student will be able to understand how to use place value and basic math to find patterns in division problems. Lesson 31 Dividing Decimals by Whole Numbers Objective: The student will be able to divide a decimal by a whole number. Lesson 32 Fraction Conversions Objective: The student will be able to convert a fraction to a decimal. Lesson 33 Decimal Conversions Objective: The student will be able to convert a decimal to a fraction.

7 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 34 Dividing Decimals by Decimals Decimal Division Assessment Objective: The student will obtain a general understanding of how to divide Decimals. Lesson 35 Working with Expressions Objective: The student will learn to work with algebraic expressions. Lesson 36 Working with Equations Objective: The student will gain additional experience working with equations. Lesson 37 Order of Operations Objective: The student will be able to solve problems utilizing the order of operations. Lesson 38 Finding Patterns Objective: The student will be able to find the missing numbers in various patterns. Lesson 39 Working with Functions Objective: The student will understand the definition of a function and be able to complete a function table by using an equation. Lesson 40 The Commutative and Associative Properties Objective: The student will be able to understand and use the commutative and associative properties of multiplication. Lesson 41 The Identity Property Objective: The student will be able to understand and use the identity property of multiplication. Lesson 42 The Distributive Property Division in Algebra Assessment Objective: The student will be able to understand and use the distributive property in multiplication. Lesson 43 Divisibility Objective: The student will be able to use Divisibility rules. Lesson 44 Prime and Composite Numbers Objective: The student will be able to identify whether a number is prime or composite.

8 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 45 Finding the Greatest Common Factor Objective: The student will be able to find the Greatest Common Factor of a pair of numbers. Lesson 46 Finding the Least Common Multiple Factors and Multiple Assessment Objective: The student will be able to find the Least Common Multiple of a pair of numbers. Lesson 47 Introduction to Exponents Objective: The student will be able to use exponential notation correctly. Lesson 48 Understanding the Expanded Form of a Number Objective: The student will be able to use place value to write the expanded form of a number. Lesson 49 Prime Factors and Exponents Introduction to Exponents Assessment Objective: The student will be able to use exponents to compose the prime factorization of a number. Lesson 50 Finding Equivalent Fractions Objective: The student will be able to create equivalent fractions Lesson 51 Reducing Fractions Objective: The student will be able to reduce fractions using the Greatest Common Factor. Lesson 52 Mixed Numbers Objective: The student will be able to convert an improper fraction to a mixed number,and a mixed number to an improper fraction. Lesson 53 Comparing and Ordering Fractions Objective: The student will be able to compare fractions with unlike denominators, and put fractions in order from least to greatest. Lesson 54 Fractions and Decimals Objective: The student will be able to convert fractions to decimals Lesson 55 Fractions and Decimals Assessment Objective: The student will learn to convert decimals to fractions.

9 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 56 Addition and Subtraction of Fractions with Like Denominators Objective: The student will be able to add and subtract fractions with like denominators Lesson 57 Addition and Subtraction of Fractions with Unlike Denominators Objective: The student will be able to add fractions with unlike denominators Lesson 58 Subtraction with Unlike Denominators Objective: The student will be able to subtract fractions with unlike denominators Lesson 59 Sums and Differences Objective: The student will be able to use estimation to figure out the sum or difference of a fraction problem where the fractions have unlike denominator Lesson 60 More Addition and Subtraction of Fractions Objective: The student will be able to add and subtract fractions by converting fractions to common denominators. Lesson 61 Addition and Subtraction of Fractions Assessment Objective: The student will practice adding and subtracting with fractions. Lesson 62 Addition of Mixed Fractions Objective: The student will be able to add mixed fractions by changing each fraction to the lowest common denominator. Lesson 63 Subtraction of Mixed Fractions Objective: The student will be able to subtract mixed fractions using the lowest common denominator. Lesson 64 More Subtraction of Mixed Fractions Objective: The student will be able to subtract mixed fractions requiring renaming. Lesson 65 Addition and Subtraction of Mixed Fractions Assessment Objective: The student will practice adding and subtracting mixed fractions. Lesson 66 Multiplication of Fractions by Fractions Objective: The student will be able to multiply a fraction by a fraction.

10 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 67 Multiplication of Fractions by Whole Numbers Objective: The student will be able to multiply a fraction by a whole number. Lesson 68 Multiplication of Fractions by Mixed Numbers Objective: The student will be able to multiply a fraction by a mixed number. Lesson 69 Multiplication of Mixed Numbers Objective: The student will be able to multiply two mixed fractions together. Lesson 70 Multiplication Review - Multiplication of Fractions Assessment Objective: The student will be able to review the multiplication of fractions. Lesson 71 Reciprocals Objective: The student will be able to find the reciprocal of a number. Lesson 72 Dividing a Fraction by a Fraction Objective: The student will be able to divide a fraction by a fraction. Lesson 73 Dividing a Fraction by a Whole Number Objective: The student will be able to divide a fraction by a whole number Lesson 74 Dividing a Fraction by a Mixed Number Objective: The student will be able to divide a fraction by a mixed number. Lesson 75 Dividing Mixed Fractions Objective: The student will be able to divide two mixed fractions. Lesson 76 Division Review - Division of Fractions Assessment Objective: The student will be able to review the various types of fraction division he has already learned. Lesson 77 Geometry Terminology Objective: The student will be able to understand and apply basic geometric Vocabulary related to lines and angles. Lesson 78 Angles Objective: The student will be able to use measuring tools to measure angles.

11 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 79 Polygons Objective: The student will be able to classify an angle and polygon by type. Lesson 80 Circles Objective: The student will be able to understand and identify the properties of circles,and use a compass to draw circles correctly given specific information Lesson 81 Congruent and Similar Figures Objective: The student will be able to identify two figures as congruent or similar. Lesson 82 Symmetry Geometry Assessment Objective: The student will be able to draw lines of symmetry into a figure, and determine whether a figure has rotational symmetry. Lesson 83 Introduction to Plane and Solid Figures Objective: The student will be able to use an artistic collage to classify plane and solid figures. Lesson 84 Triangles Objective: The student will be able to classify triangles. Lesson 85 Quadrilaterals Objective: The student will be able to classify quadrilaterals. Lesson 86 Transformations Objective: The student will be able to understand and use the various types of transformations. Lesson 87 Plane and Solid Figures Assessment Objective: The student will be able to identify solid figures. Lesson 88 Graphing Introduction Objective: The student will be able to use an X-Y table to graph on the coordinate plane. Lesson 89 Graphing on a Coordinate Plane Objective: The student will

12 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 90 Integers Introduction Objective: The student will be able to understand the number line and find the absolute value of a number. Lesson 91 Comparing Integers Objective: The student will be able to compare integers using the greater than, less than, or equal to symbols. The student will also be able to place integers in order from least to greatest. Lesson 92 Operations with Integers Part 1 Objective: The student will be able to add integers. Lesson 93 Operations with Integers Part 2 Objective: The student will be able to subtract two integers. Lesson 94 Coordinate Plane and Integers Review Objective: The student will be able to review the skills he has learned in preparation for the Assessment given on the coordinate plane and integers. Lesson 95 Coordinate Plane and Integers Assessment Objective: The student will be able to display her understanding of Coordinate Planes and Integers by taking the Coordinate Plane and Integers Assessment. Lesson 96 Length Objective: The student will be able to understand length in terms of the metric system. Lesson 97 Length Activity Objective: The student will be able to understand the metric system as it relates to measuring length. Lesson 98 Metric Conversions Objective: The student will be able to make conversions using metric units of length. Lesson 99 Customary Conversions Objective: The student will be able to make conversions using customary units of Length. Lesson 100 Capacity and Weight Objective: The student will be able to make conversions using customary units of capacity and weight. Lesson 101 Capacity and Mass Objective: The student will be able to make conversions using metric units of capacity and mass.

13 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 102 Time Objective: The student will be able to determine elapsed time and make conversions using units of time. Lesson 103 Temperature Objective: The student will be able to convert between Fahrenheit and Celsius. Lesson 104 The Metric System Review Objective: The student will be able to review the Metric System. Lesson 105 The Metric System Assessment Objective: The student will be able to display knowledge about the Metric System. Lesson 106 Measuring Perimeter Objective: The student will be able to use a ruler to measure the perimeter of various objects in the real world. Lesson 107 Finding Perimeter Objective: The student will be able to find the perimeter of an object when she is not given all of the lengths of the object. Lesson 108 Finding Circumference Objective: The student will be able to find the circumference of a circle. Lesson 109 Circumference and Diameter Objective: The student will be able to understand the relationship between circumference and diameter. Lesson 110 Perimeter Review Objective: The student will review finding perimeter and circumference in preparation for an Assessment. Lesson 111 Perimeter Assessment Objective: The student will be able to display his knowledge of finding perimeter and circumference by completing the Perimeter Assessment. Lesson 112 What is Area? Objective: The student will be able to use grid paper to estimate area.

14 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 113 Finding Area Part 1 Objective: The student will be able to find the area of a rectangle or square. Lesson 114 Area and Perimeter Objective: The student will be able to practice finding area and perimeter of squares and rectangles. Lesson 115 Finding Area Part 2 Objective: The student will be able to find the area of a triangle. Lesson 116 Finding Area Part 3 Objective: The student will be able to find the area of a parallelogram. Lesson 117 Area Review Objective: The student will practice finding the area of rectangles, squares, triangles, and parallelograms. Lesson 118 Area Assessment Objective: The student will complete the Area Assessment Lesson 119 Million Dollar Home Project Part 1 Objective: The student will be able to use knowledge of area and perimeter to design a million-dollar home. Lesson 120 Million Dollar Home Project Part 2 Objective: The student will be able to use knowledge of area and perimeter to design a home if he was given one million dollars. Lesson 121 Million Dollar Home Project Part 3 Objective: The student will determine the scale for her dream-home drawing, and will calculate total area and perimeter of the lot and floor plan. Lesson 122 Million Dollar Home Project Part 4 Objective: The student will pick out furnishings and begin budgeting his million-dollar home. Lesson 123 Million Dollar Home Project Part 5 Objective: The student will calculate the average square footage and cost-to-furnish of her million dollar home. She will also answer questions she will need for the presentation she will give in Lesson 124.

15 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 124 Million Dollar Home Project Presentation Objective: The student will orally present his Million Dollar Home Project Lesson 125 Surface Area Objective: The student will be able to find the surface area of a rectangular prism. Lesson 126 Finding Volume Part 1 Objective: The student will be able to find the volume of a rectangular prism. Lesson 127 Finding Volume Part 2 Objective: The student will be able to find the volume of a rectangular prism when given only two of the three determining measurements. Lesson 128 Perimeter, Area, and Volume Objective: The student will be able to find the surface area of a rectangular prism. Lesson 129 Volume Review Objective: The student will be able to find the volume of various figures using the appropriate measurement tool. Lesson 130 Perimeter, Area, and Volume Assessment Objective: The student will be able to find the surface area and volume of a solid figure. Lesson 131 What is a Ratio? Objective: The student will be able to understand what a ratio is. Lesson 132 Writing Ratios Objective: The student will be able to write ratios. Lesson 133 Proportions Objective: The student will be able to determine if two ratios are equivalent. Lesson 134 More Proportions Objective: The student will be able to determine if two ratios are equivalent in order to solve proportions. Lesson 135 Solving Proportions Objective: The student will be able to determine the missing value in a proportion.

16 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 136 Ratios Review Objective: The student will be able to work with ratios and proportions. Lesson 137 Ratios Assessment Objective: The student will be able to display her knowledge of ratios and proportions on the Assessment for this chapter. Lesson 138 Scale Drawing U.S. Map Objective: The student will be able to use proportions to determine distances in scale drawings. Lesson 139 Scale Drawing Workspace Objective: The student will be able to create a scale drawing. Lesson 140 Scale Drawing Project Living Room Objective: The student will be able to use proportions to construct a scale drawing of his living room. Lesson 141 Scale Drawing Project Kitchen Objective: The student will be able to use proportions to construct a scale drawing. Lesson 142 Scale Drawing Project Back Yard Objective: The student will be able to use proportions to construct a scale drawing. Lesson 143 Scale Drawing Project Shrinking it Down Objective: The student will be able to use proportions to construct a scale drawing. Lesson 144 Understanding Percentages Objective: The student will be able to understand how to write a percentage. Lesson 145 Changing Decimals to Percentages Objective: The student will be able to convert a decimal to a percentage. Lesson 146 Changing Percentages to Decimals Objective: The student will be able to convert a percentage to a decimal. Lesson 147 Changing Fractions to Percentages Objective: The student will be able to understand how to convert a fraction to a percentage. Lesson 148 Changing Percentages to Fractions Objective: The student will be able to convert a percentage to a fraction.

17 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 149 Fractions, Decimals, and Percentages Objective: The student will be able to make conversions between fractions, decimals, and percentages. Lesson 150 Comparing Fractions, Decimals, and Percentages Objective: The student will be able to make comparisons between fractions, decimals,and percentages. Lesson 151 Finding a Percentage of a Number Objective: The student will be able to find a requested percentage of a number. Lesson 152 Percentage Review Objective: The student will review percentage computations. Lesson 153 Percentages Assessment Objective: The student will complete an Assessment covering percentages. Lesson 154 Permutations Objective: The student will see how to determine the number of permutations possible in a given set. Lesson 155 Combinations Objective: The student will be able to arrange numbers and objects into combinations. Lesson 156 The Fundamental Counting Principle Objective: The student will become familiar with the fundamental counting principle Lesson 157 Combinations, Permutations, and Counting Review Objective: The student will gain additional practice building combinations and permutations and in using the fundamental counting principle before moving on to a study of probability and statistics. Lesson 158 Introduction to Probability Objective: The student will be able to understand what probability is and how it is used. Lesson 159 Probability Objective: The student will be able to understand and apply the basic concepts of Probability Lesson 160 Probability Activity Objective: The student will be able to use probability activities to determine if a game is fair.

18 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 161 Understanding Probability and Odds Objective: The student will use a combination of pattern making, counting, and real world experimentation to better understand probability and odds. Lesson 162 Writing Probability Objective: The student will be able to use fractions to write probabilities. Lesson 163 Probability and Counting Review Objective: The student will complete a review of probability and combinations. Lesson 164 Probability and Counting Assessment Objective: The student will be able to display all knowledge learned by completing the Probability Assessment. Lesson 165 Mean Objective: The student will be able to determine the mean of a set of data. Lesson 166 Median Objective: The student will be able to find the median of a set of data. Lesson 167 Mode Objective: The student will be able to find the mode of a set of data. Lesson 168 Frequency Table Objective: The student will be able to make a frequency table in order to analyze data. Lesson 169 Line Plot Objective: The student will be able to represent data by creating a line plot. Lesson 170 Stem and Leaf Plot Objective: The student will be able to create a stem and leaf plot to represent data. Lesson 171 Pie Chart Objective: The student will be able to create a pie chart to display data. Lesson 172 Bar Graph Objective: The student will be able to represent data by creating a bar graph.

19 TABLE OF CONTENTS AND OBJECTIVES FIFTH GRADE MATH Lesson 173 Data Analysis Review Objective: The student will study for an Assessment over data analysis techniques. Lesson 174 Data Analysis Assessment Objective: The student will be able to display his knowledge and understanding of data analysis. Lesson 175 Culminating Project Part 1 Objective: The student will demonstrate her knowledge of 5th grade math by composing an Assessment. Lesson 176 Culminating Project Part 2 Objective: The student will continue to work on his culminating project. Lesson 177 Culminating Project Part 3 Objective: The student will continue to work on her culminating project. Lesson 178 Culminating Project Part 4 Objective: The student will review his questions so far, and add more in this final drafting Lesson. Lesson 179 Culminating Project Part 5 Objective: The student will rewrite her Assessment to make sure it is neat, legible, and presentable. Lesson 180 Culminating Project Completion Objective: The student will administer and / or take the 5th grade Math final Assessment

20 Fifth Grade Math Common Core Alignment Complete Curriculum Lesson Operations and Algebraic Thinking --Write and interpret numerical expressions. 5.0A.1 Use parentheses, brackets, or braces in numerical 13, 37, 40, 42, 88 expressions, and evaluate expressions with these symbols. 5.0A.2 Write simple expressions that record calculations with 37, 40, 42 numbers, and interpret numerical expressions without evaluating them. For example, express the calculation add 8 and 7, then multiply by 2 as 2 (8 + 7). Recognize that 3 ( ) is three times as large as , without having to calculate the indicated sum or product. --Analyze patterns and relationships. 5.OA.3 Generate two numerical patterns using two given rules. *39 Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule Add 3 and the starting number 0, and given the rule Add 6 and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. Number and Operations in Base Ten --Understand the place value system. 5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 5, 6, 7, 21, 48 5.NBT.2 Explain patterns in the number of zeros of the product 48 when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of NBT.3 Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten 5, 55, 149 numerals, number names, and expanded form, e.g., = (1/10) + 9 (1/100) + 2 (1/1000). b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. 7 5.NBT.4 Use place value understanding to round decimals to any 9 place. --Perform operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.5 Fluently multiply py multi-digit whole numbers using the 18, 19, 20 standard algorithm.

21 5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Number and Operations Fractions --Use equivalent fractions as a strategy to add and subtract fractions , 21-24, 30, 31, 34, NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/ /12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) 5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. 56, 61 --Apply and extend previous understandings of multiplication and division to multiply and divide fractions. py 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a b). Solve word problems involving division of whole numbers leading to answers in the form of fractions, mixed numbers, or decimal fractions, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a q b. For example, use a visual fraction model to show (2/3) 4 = 8/3, and create a story context for this equation. Do the same with (2/3) (4/5) = 8/15. (In general, (a/b) (c/d) = ac/bd.) 66-70, 73

22 b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. *112 5.NF.5 Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n a)/(n b) to the effect of multiplying a/b by 1. 19, 20, , 76 5.NF.6 Solve real world problems involving multiplication of 67, 68 fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, 73, 74, 76 and compute such quotients. For example, create a story context for (1/3) 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) 4 = 1/12 because (1/12) 4 = 1/3. b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 (1/5) = 20 because 20 (1/5) = 4.

23 c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? 73, 76 Measurement and Data --Convert like measurement units within a given measurement system. 5.MD.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. --Represent and interpret data. 5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. --Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition , *169 5.MD.3 Recognize volume as an attribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a unit cube, is said to have one cubic unit of volume, and can be used to measure volume. b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units , , MD.4 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 5.0 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. 126,

24 b. Apply the formulas V = l w h and V = b h for rectangular prisms to find volumes of right rectangular prisms with wholenumber edge lengths in the context of solving real world and mathematical problems. 126, 127, 129, 130 c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. 126, 128 Geometry --Graph points on the coordinate plane to solve real-world and mathematical problems. 5.G.1 Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y- axis and y-coordinate). 88, 94, 95 5.G.2 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. --Classify two-dimensional figures into categories based on their properties. 5.G.3 Understand that attributes belonging to a category of twodimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. 88, 89, 94, 95 79, 84, 85, G.4 Classify two-dimensional figures in a hierarchy based on 77, 79, 84, 85, 176 properties.

25 STUDENT MANUAL MATH LESSON 1-1 Lesson 1 Introduction to Place Value whole number and decimal concepts. Read this Place Value Chart below present. The chart shows the place value of the number 128,500. HUNDRED Thousands TEN Thousands Thousands Hundreds Tens Ones x 100,000 2 x 10,000 8 x 1,000 5 x x 10 0 x 1 100,000 20,000 8, chart above. The 1 in the number is in the hundred thousands place; the 2 in the number is in the ten thousands place; the 8 in the number is in the in the tens place; the next 0 is in the ones place. Work through these Examples with your teacher. To determine the place value of the bold digit for each problem , , , , 200

26 STUDENT MANUAL MATH LESSON 1-2 To write a number in expanded form, use each digit, multiplied by the place value of the digit, and add each of these products together. Example: Write 14,600 in expanded form 1 x 10, x 1, x x x 1 10, , Example: Write 235,781 in expanded form Write 45 in expanded form To write a number using word form, simply write the words represented by the place value of each number, separated by commas. Example: Write 481 in word form Example: Write 12,300 in word form Example: Write 132,450 in word form Lesson Wrap-Up: Look at the addresses on houses as you walk or ride past them and identify the place value of each number in each address.

27 STUDENT MANUAL MATH LESSON 1-3 Lesson 1 Practice Sheet Part 1: Write the place value of the digit in bold text , , , , , , , , , , , , , , , , , , , ,021 Part 2: Write each number in expanded form , , ,

28 STUDENT MANUAL MATH LESSON 1-4 Part 3: Write each number in word form. 1. 1, , , , , , , , , , , , ,387

29 STUDENT MANUAL MATH LESSON 2-1 Lesson 2 Understanding Millions and Billions In Lesson 1, we looked at the place value of numbers that are not any greater than a number in the hundred thousands. Now you will become familiar with numbers in the millions and billions. Look at the Place Value Chart below which has millions and billions added to it for the number 134,567,302,119. Lesson 2 Place Value Chart Hundred Billions Ten Billions Billions Hundred Thousands Ten Thousands Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones x 100,000,000,000 3 x 30,000,000,000 4 x 1,000,000,000 5 x 100,000, ,000,000,000 90,000,000,000 4,000,000, ,000,000 60,000,000 7,000, ,000 00,000 2, x 10,000,000 7 x 1,000,000 3 x 100,000 0 x 10,000 2 x 1,000 1 x x 10 9 x 1

30 STUDENT MANUAL MATH LESSON 2-2 Therefore, the expanded form of the number 134,567,302,119 as we learned in the previous Lesson is: 100,000,000, ,000,000, ,000,000, ,000, ,000, ,000, , , The word form of the number above as we learned in the previous Lesson is: three hundred two thousand, one hundred nineteen. Do the following with each Example: Tell the place value of the bold digit. Write the number in expanded form. Write the number in word form. Example: 2,254,380 Example: 43,076,987,100 Example: 88,330,210 Lesson Wrap-Up: How many digits are there in a million? A billion?

31 STUDENT MANUAL MATH LESSON 2-3 Lesson 2 Practice Sheet Part 1: Write the place value of the bold digit. 1. 2,432, ,410, ,682,117, ,252, ,240, , ,890,763, , , , , ,456, ,035, ,234,888, , ,742,445, ,485, ,321, ,190, ,245,038, ,523, ,098,467, ,021,190

32 STUDENT MANUAL MATH LESSON 2-4 Part 2: Write each number in expanded form ,975, ,566,788, ,361,455, ,301, , ,116, , , ,284, , , ,677, , ,387

33 STUDENT MANUAL MATH LESSON 2-5 Part 3: Write each number in word form ,949, ,529,902, , ,420,100, , ,865,132, ,455,679, ,520, ,763, ,345, , ,984,975,900

34 STUDENT MANUAL MATH LESSON 3-2 Lesson 3 Important Whole Numbers Benchmark number: Practice: Step 1: Step 2: Step 3: Lesson Wrap-Up: PRINT

35 STUDENT MANUAL MATH LESSON 4-1 Lesson 4 Comparing and Ordering Whole Numbers Whole umbers N Assessment In this Lesson, we will practice comparing and putting whole numbers in order. The easiest process to use when doing this is to examine the place value of the numbers. Let s compare whole numbers with an Example: Example: Compare the number 81,928 and 81,968 When two digits that differ are found, determine which digit is larger. 81,928 vs. 81,968 Complete these Practice Problems to review comparing numbers: 34,569 3,450 1,230,189 1,330, , ,510 88,914,320 89,391,020

36 STUDENT MANUAL MATH LESSON 4-2 Complete these Examples of ordering whole numbers: Put the following numbers in order from least to greatest: 37,430 37,510 37,100 The first step when ordering whole numbers is to compare each number using place value as we did previously in this Lesson. Next, determine the smallest number in the set and place this number at the beginning of your new list. Repeat these steps as needed. Complete the following Practice Problems to review ordering of whole numbers ,280,438 1,281,567 1,284, ,458 10,451 10, , , ,120 Lesson Wrap-Up: Complete the Lesson 4 Practice Sheet. When you are finished, your teacher will assign the Whole Numbers Assessment.

37 STUDENT MANUAL MATH LESSON 4-3 Lesson 4 Practice Sheet Part 1: Compare each set of numbers using <, >, = ,120 82, ,165 7, , , ,263 6, ,431 7, ,234 5, ,689 6, , , , ,139

38 STUDENT MANUAL MATH LESSON 4-4 Part 2: Put each set of numbers in order from least to greatest ,250, 1,298, 1,103, 1,192, 1,153, 1, , 966, 962, 967, 984, 992, , ,810, 879,872, 815, , 117, ,746, 23,786, 23,709, 23, , 251, 254, 236, ,419, 3,206, 3,246, 3,525 3, , 840, 894, , 702, 712, 751, 760

39 STUDENT MANUAL MATH LESSON 4-5 Whole Numbers Assessment Part 1: For each number, tell the place value of the digit in bold, write the number in expanded form, and write the number in word form. 1. 8,456 Value: Expanded: Word: 2. 8,221,098 Value: Expanded: Word: 3. 76,546,890 Value: Expanded: Word: ,480 Value: Expanded: Word:

40 STUDENT MANUAL MATH LESSON ,842 Value: Expanded: Word: 6. 40,634,123 Value: Expanded: Word: ,889,912,310 Value: Expanded: Word: Value: Expanded: Word:

41 STUDENT MANUAL MATH LESSON ,256 Value: Expanded: Word: ,000,437 Value: Expanded: Word: 11. 7,551,191 Value: Expanded: Word: ,388 Value: Expanded: Word:

42 STUDENT MANUAL MATH LESSON , 841 Value: Expanded: Word: ,789,100 Value: Expanded: Word: ,976 Value: Expanded: Word: ,077 Value: Expanded: Word:

43 STUDENT MANUAL MATH LESSON 4-9 Part 2: Compare each set of numbers using <, >, = ,430 42, ,731 7, ,018 7, ,444,310 4,444, , , ,998 45, ,112 26,211 Part 3: Put each set of numbers in order from least to greatest , , ,709, 777,770, 746, ,209,251, 1,254,236, 1,209,208, 1,238, ,345, 1,306, 1,346, 1,325, 1,321, 1, , 845, 814, 893, ,128, 763,102, 763,172, 736,121, 736,106

44 STUDENT MANUAL MATH LESSON 5-1 Lesson 5 Introduction to Decimal Place Value The numbers we use every day are part of the decimal system. In order of the digits in a number. A Place Value Chart can be a helpful tool when determining the value of a number, and in order to compare decimals, it is necessary to have a solid knowledge base in place value. Examine the Decimal Place Value Sequence below: Ones Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths Learn how to read and write numbers using place value through the Examples below. Example: when placed in a decimal value chart looks like this: Ones Tenths Hundredths Thousandths In expanded form, looks like this: And in written form, looks like: Five and forty-nine thousandths.

45 STUDENT MANUAL MATH LESSON 5-2 Example: In a Place Value Chart, looks like: Ones Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths In expanded form, looks like this: And in written form, looks like: Now, go through these examples with your teacher.

46 STUDENT MANUAL MATH LESSON 5-3 Complete the following problems to practice using a Place Value Chart to represent a decimal, and write the decimal in the expanded and written forms Tens Ones Tenths Hundredths Ones Tenths Hundredths Thousandths 3. 1, Thousands Hundreds Tens Ones Tenths Hundredths Thousandths Lesson Wrap-Up: Complete the Lesson 5 Practice Sheet.

47 STUDENT MANUAL MATH LESSON 5-4 Lesson 5 Practice Sheet For each decimal given, write the decimal in expanded form, and write the decimal in word form Expanded: Word: Expanded: Word: Expanded: Word: 4. 5, Expanded: Word: Expanded: Word:

48 STUDENT MANUAL MATH LESSON Expanded: Word: Expanded: Word: Expanded: Word: Expanded: Word: Expanded: Word:

49 STUDENT MANUAL MATH LESSON Expanded: Word: Expanded: Word: 13. 1, Expanded: Word: Expanded: Word: Expanded: Word:

50 STUDENT MANUAL MATH LESSON Expanded: Word: Expanded: Word: Expanded: Word: Expanded: Word: Expanded: Word:

51 STUDENT MANUAL MATH LESSON Expanded: Word: Expanded: Word: , Expanded: Word: Expanded: Word: Expanded: Word:

52 STUDENT MANUAL MATH LESSON 6-1 Lesson 6 Finding Decimal Equivalents It is important, when working with decimals, to remember that there is often more than one way to write a decimal. An equivalent decimal is a decimal that has a different name for the same amount. The easiest way to compare two decimals to determine whether they are equivalent is to line the two decimals up and compare each digit in the same place-value position, working from the left to the right. It is also important to note that if there is a zero as the last digit in a decimal, the zero is not needed and can be eliminated without changing the value of the decimal. Complete the following problems to determine whether the decimal pairs are equivalent or not and and and and and Lesson Wrap-Up: Write 20 pairs of decimals and trade them with a partner to practice determining whether the pairs are equivalent or not.

53 STUDENT MANUAL MATH LESSON 7-1 Lesson 7 Ordering and Comparing Decimals Decimals Assessment The numbers we use every day are part of the decimal system. In order to find the value of a number, it is helpful to look at the digits and position of the digits in a number. A Place Value Chart can be a helpful tool when determining the value of a number, and in order to compare decimals, it is necessary to have a solid knowledge base in place value. Let s examine the Place Value Chart below: Here s how to compare and order decimals through several Examples: Example: Sally is asked to compare and order the following weights: 12.2, 12.7, 11.6 Step 1: Sally compares the first two numbers. Both numbers have a 1 in the tens place, so Sally must then look at the numbers in the ones place. Both numbers have a 2 in the ones place, so Sally must now look at the tenths place. The first number has a 2 in the tenths place and the second number has a 7 in the tenths place. Since 2 is less than 7, 12.2 < Step 2: Sally now compares the third number, 11.6, to the greatest number so far, Both numbers have a 1 in the tens place, so Sally must look at the numbers in the ones place. The first number has a 1 in the ones place, and the second number has a 2 and since 2 is greater than 1, 12.7 > Step 3: Therefore, the ordered list of decimals from least to greatest is 11.6, 12.2, and then 12.7.

54 STUDENT MANUAL MATH LESSON 7-2 Example: John found three different prices for monthly phone service rates including $19.99, $20.50, and $ Order the rates from least to greatest and find the lowest rate. You should be able to mentally add and subtract 0.1, 0.01 and to a decimal. You need to look at the number in the same decimal place as the 1 and add 1 to the number or subtract 1 from the number. Add these numbers, mentally: = = = Subtract these numbers, mentally: = = = Lesson Wrap-Up: Complete the Lesson 7 Practice.

55 STUDENT MANUAL MATH LESSON 7-3 Lesson 7 Practice Sheet Part 1: Compare each set of decimals

56 STUDENT MANUAL MATH LESSON 7-4 Part 2: Order the decimals from least to greatest , 0.32, 0.12, , 0.587, 0.593, , 0.131, 0.33, 0.3

57 STUDENT MANUAL MATH LESSON 7-6 Decimals Assessment Part 1: For each decimal given, write the decimal in expanded form and in word form Expanded: Word: Expanded: Word: Expanded: Word: 4. 3,126.9 Expanded: Word: Expanded: Word: Expanded: Word:

58 STUDENT MANUAL MATH LESSON 7-7 Part 2: For each pair of decimals, determine whether the two decimals are equivalent or not equivalent and and and and and and 3.24 Part 3: Add or subtract the decimals, mentally = = = = = = = = = =

59 STUDENT MANUAL MATH LESSON 7-8 Part 4: Order each set of decimals from least to greatest , 0.12, 0.36, 0.41, , 12.03, 12.75, 12.12, 12.34, , , , , , PRINT

60 STUDENT MANUAL MATH LESSON 8-1 Lesson 8 Rounding Whole Numbers It is sometimes helpful to understand how to round whole numbers in order to solve problems with real-life applications. In this Lesson, you will learn how. Examine the real-life problem below to understand how rounding is helpful in this particular case: Problem: The average attendance at the Dragons football games is 1,356 fans on a given Friday night. It is expected that most fans will purchase a hot dog from the concession stand at some point during the game. If the hot dogs come in boxes of 100, how many boxes should be purchased and prepared? Solution: We will round the number of fans attending the football game up from 1,356 to 1,400. If the hot dogs come in boxes of 100, then 14 boxes should be purchased and prepared because 14 x 100 = 1,400. In instances, such as the one stated in the above problem, it is usually better to overestimate than underestimate, as you would not want to run out of food to feed a crowd. Let s look at the process involved in rounding whole numbers: Determine the place to which you are rounding (in many cases, this will be given to you). 5 If the digit to the right of this place is 5 or greater, you will round up.

61 STUDENT MANUAL MATH LESSON 8-2 Examine the following Examples of rounding with whole numbers. Round 1,343 to the nearest hundred Round 10,890 to the nearest thousand Round 127 to the nearest ten Round 1,356,280 to the nearest million Lesson Wrap-Up: Complete the Lesson 8 Practice Sheet.

62 STUDENT MANUAL MATH LESSON 8-3 Lesson 8 Practice Sheet Directions: Round each number to the place value of the digit in bold. 1. 2, , , , , , , , , , , , , ,223, , ,107

63 STUDENT MANUAL MATH LESSON 9-1 Lesson 9 Rounding Decimals As you learned in the previous Lesson, rounding is a simple process if you follow a specific procedure. In this Lesson, you will use that process to practice rounding decimals in the same manner that we used to round whole numbers. Let s review the process that we learned in the last Lesson. Determine the place to which you are rounding (in many cases, this will be given to you). If the digit to the right of this place is less than 5, you will round down. If the digit to the right of this place is 5 or greater, you will round up. Let s examine the following and round each decimal to a given place value. Example: Round to the nearest hundredth Example: Round to the nearest tenth Example: Round to the nearest hundredth Example: Round to the nearest thousandth Example: Round to the nearest whole number.

64 STUDENT MANUAL MATH LESSON 9-2 Lesson 9 Practice Sheet Round each decimal to the place value of the bold digit. PRINT

65 STUDENT MANUAL MATH LESSON 10-1 Lesson 10 Estimating As you learned in Lesson 8, it is often easy and helpful to solve word problems and real-life application problems by estimating and rounding. In this Lesson, you will practice using rounding to estimate the answer to addition and subtraction problems. Let s start by reviewing rounding with the following Examples: Practice rounding each number to the nearest thousand: 1. 3, , , ,341 Practice adding or subtracting by rounding to the nearest hundred: ,530 1, , ,125

66 STUDENT MANUAL MATH LESSON 10-2 Now, we will look at estimating sums and differences with decimals. Practice rounding each decimal to the nearest tenth nearest whole number: Complete the following Practice Problems by using estimation to figure out each sum or difference. 1. 1, ,489 (round to nearest hundred) (round to nearest ten) (round to nearest whole number) (round to nearest whole number) Lesson Wrap-Up: Complete the Lesson 10 Practice Sheet.

67 STUDENT MANUAL MATH LESSON 10-3 Lesson 10 Practice Sheet Estimate each sum or difference. Round to the Nearest Tens 1. 2, , Nearest Hundreds , , , , ,580 Nearest Tens Nearest Hundreds 19. 6,177-5, ,471-1,

68 STUDENT MANUAL MATH LESSON 11-1 Lesson 11 Addition and Subtraction of Whole Numbers In Lesson 10, you learned that estimating is one way to determine a sum or a difference. Although you will not usually be asked to estimate an answer to an addition or subtraction problem, it is a good technique to use if you want to check your answer to a problem. In each of this Lesson s problems, use the estimation technique you learned in Lesson 10 to check your answer. Practice adding and subtracting with the following Examples (Don t forget to check your answers by estimating to the nearest thousand): Example: 1, ,894 Example: 92, ,480 Example: 4,965-3,136 Lesson Wrap-Up: Complete the Lesson 11 Practice Sheet.

69 STUDENT MANUAL MATH LESSON 11-2 Lesson 11 Practice Sheet Add or subtract Add 1. 1, , , , , , , , , , , , ,599 Subtract 17. 1, , , , ,

70 STUDENT MANUAL MATH LESSON 12-1 Lesson 12 Addition and Subtraction with Decimals Addition and Subtraction with Decimals Assessment Think about the times in everyday life when you have to add or subtract decimals. When you are buying anything, most prices are in decimal form. For Example, Johnny is buying candy at his local store, and he wants to buy a box of chocolate for $2.40, a sucker for $0.80, and a bag of hard candy for $1.35. In order to determine the total cost of his purchase, Johnny will need to add all three prices that hav decimals. It is important when adding or subtracting decimals to align the decimal points in each number, and then to place the decimal point in the answer directly below the decimal point in the last number in your problem. Example Add To add these numbers you can change them to fractions. Add the fractions and change them back to decimals. You would add 10 6/ /10 = 25 8/10 = 25.8 Practice = = Add / /100 = 55 63/100 = Practice = =

71 STUDENT MANUAL MATH LESSON 12-2 Practice adding with decimals: Example: Add Examples: 1. Add Add Add Example To subtract these numbers you can change them to fractions. Subtract the fractions and change them back to decimals. Subtract / /10 = 15 3/10 = 15.3 Practice = = Subtract = 78 45/ /100 = 56 11/100 = Practice = =

72 STUDENT MANUAL MATH LESSON 12-3 Example: Subtract Examples: 1. Subtract Subtract Subtract Lessons Wrap-up: Complete the Lesson 12 Practice Sheet.

73 STUDENT MANUAL MATH LESSON 12-4 Adding and Subtracting with Decimals Practice Find each sum or difference = = = = = =

74 STUDENT MANUAL MATH LESSON = = = = = =

75 STUDENT MANUAL MATH LESSON 12-6 Adding and Subtracting with Decimals Assessment Part 1: Round each number to the given place value 1. 12,589 to the nearest hundred 2. 8,456,123 to the nearest ten thousand 3. 3,471 to the nearest thousand to the nearest hundredth to the nearest tenth to the nearest thousandth to the nearest whole number 8. 3, to the nearest thousand to the nearest hundredth to the nearest thousandth

76 STUDENT MANUAL MATH LESSON 12-7 Part 2: Solve each addition or subtraction problem. Then, check your answer using estimation ,973-9, , , PRINT

77 STUDENT MANUAL MATH LESSON 13-1 Lesson 13 Introduction to Expressions and Variables Now, we begin learning the basics of algebra, utilizing what you have learned thus far about addition and subtraction. be used in this Lesson: Expression: An algebraic expression is a mathematical phrase or sentence. Variable: A variable is a letter that represents a number. Some Examples of expressions can be seen below: Example: Example: 3X + 2Y Example: 1(4X 3) Example: Note that the apparent difference between an expression and an equation is an equal sign. An equation includes an equal sign an expression does not. including word problems. Look at the following Example: the second half. To represent the total points the team scored, you can use an equation: Total points scored = If you did not know how many points the football team scored in the second half, you could use a variable to represent the points. You can use any letter of the alphabet as a variable, although it is common to use a letter that corresponds to the piece of information it will represent. For example, to represent the total points scored in the second half of the football game we will use P. The algebraic expression representing the total points scored in the game would be: 21 + P.

78 STUDENT MANUAL MATH LESSON 13-2 When writing an expression, it is helpful to understand all of the various words and phrases that indicate addition, subtraction, multiplication, and division: Addition Subtraction Multiplication Division Sum Difference Product Quotient Increase Decrease Factors Equally divided More than Less than Twice Divided by Plus Minus Times Greater than Fewer than Multiplied by There are many different ways to represent multiplication in an expression. For example: 7 x B could be written in the following ways: 7B, 7(B), B x 7, or 7 B. Complete these Examples: Write a numerical expression for each word expression: 1. $9.00 less than $ Q divided by S 3. Three times a distance, D 4. The number of slices of 2 dozen apples, each of which has 5 slices

79 STUDENT MANUAL MATH LESSON 13-3 Evaluate each expression: 1. X for X = G / 10 for G = R 6 for R = 3 4. A 2B for A = 8 and B = 4 Lesson Wrap-Up: Complete the Lesson 13 Practice Sheet and review it

80 STUDENT MANUAL MATH LESSON 13-4 Lesson 13 Practice Sheet Part 1: Evaluate each expression for the given variable. 1. m - 3 for m = n for n = 5 3. r - 4 for r = x 5. d w + 27 for x = 8 for d = 6 for w = 4 7. q 4 for q = t for t = h for h = k - 52 for k = a + 42 for a = u + 17 for u = b - 2 for b = c - 10 for c = y + 41 for y = 5

81 STUDENT MANUAL MATH LESSON 13-5 Part 2: Write the expression represented. 1. a number u times 5 2. a number y less plus a number t 4. difference of 47 and a number k divided into a number x 6. a number w minus a number r less than take away 44 from a number b 9. a number n increased by a number v multiplied by 12

82 STUDENT MANUAL MATH LESSON 14-1 Lesson 14 Learning How to Write Equations What is the difference between an expression and an equation? Equation A statement showing that two quantities are equal. The difference between an expression and an equation is that an equation contains an equal sign. Examples of equations: = 30 4 X = 1 2D = 18 When an equation contains a variable, the equation will have a solution that makes the equation true. Use mental math to determine which of the following numbers is a solution to each equation X = 144 Choose: 11, 12, Y = 9 Choose: 3, 4, / A = 7 Choose: 7, 8, = Y + 15 Choose: 5, 6, 7 In order to solve word problems, you will need to understand the steps for writing and solving an equation: Step 1: Create an equation that represents the word problem. You will need to assign a variable to the unknown value in the problem. You can use any letter you would like. Step 2: Determine the correct operation sign to use in the equation. For various ways to say add", subtract", multiply". or divide". Step 3: Use mental math to determine the solution to the problem.

83 STUDENT MANUAL MATH LESSON 14-2 Examine the following Examples: Example: Tracy had 13 DVDs and got more for her birthday. This gave her a total of 25 DVDs. How many did she receive for her birthday? DVDs Example: Anna needs 18 pairs of scissors out on the tables for her students to use. She already had 12 pairs put out. How many more does she need to put out? pairs

84 STUDENT MANUAL MATH LESSON 14-3 Example: The equation W + 8 = 21 represents the number of women and men on a train. If W represents the number of women, how many women are on the train? women Example: The Statue of Liberty s hand is about 16 feet long. If the her hand? feet Lesson Wrap-Up: Write a paragraph explaining the process you would use to solve the equation 6X = 60 mentally. Then, complete the Lesson 14 Practice sheet.

85 STUDENT MANUAL MATH LESSON 14-4 Lesson 14 Practice Sheet Part 1: Solve each equation using mental math b = 48 b = = x + 85 x = 3. g + 94 = 176 g = 4. 1 = n - 8 n = g = 4 g = = d - 39 d = 7. j j = = = 7x x = 9. 8 = s 12 s = t = 247 t = e = 95 e = = b 2 b =

86 STUDENT MANUAL MATH LESSON 14-5 Part 2: Solve each word problem 13. Judy and John together own 134 marbles. Judy owns 22 more than John. Which equation should you use to find out how many John owns? A) 2m + 22 = 134 B) m = 22 C) 22 + m = 134 D) 22m = m 14. Judy and John together own 134 marbles. Judy owns 22 more than John. How many marbles does John own? A) 50 marbles B) 56 marbles C) 62 marbles D) 74 marbles 15. It costs $20.00 to rent skis. The first day s ski lift ticket costs $30.00, and for each additional day the ticket costs $7.00. How much did it cost Adam to rent skis and ski for 3 days? A) $70.00 B) $64.00 C) $80.00 D) $84.00

87 STUDENT MANUAL MATH LESSON Jim and Katie traveled 10 miles altogether. Jim traveled 2 more miles than Katie. How far did Katie travel? A) 10 miles B) 8 miles C) 6 miles D) 4 miles 17. The Mackenzie sisters spent $ on a shopping trip. If Beth spent 43 dollars more than Sarah, how much did Sarah spend? A) $60.00 B) $61.00 C) $70.00 D) $ Lauren scored 5 more points than Matt. Which of the following represents the relationship between their scores? A) L + 5 = M B) L = M + 5 C) 5L = M D) L = 5M

88 STUDENT MANUAL MATH LESSON 15-2 Lesson 15 Learning How to Solve Equations In Lesson 14, you practiced how to solve equations using mental math. In this Lesson you will continue to practice this concept. Examine the following Examples: Example: Is 10, 11, or 12 a solution to the equation 3 + x = 13? Example: Is 5, 6, or 7 a solution to the equation 20 x = 13 Practice using mental math to solve each of the following equations: X = Y = X = A 10 = B + 3 = 103 Lesson Wrap-Up: Complete the Lesson 15 Practice Sheet.

89 STUDENT MANUAL MATH LESSON 15-3 Lesson 15 Practice Sheet Solve each equation using mental math x = 73 x = = k + 70 k = 3. r + 23 = 56 r = = 95 - d 5. 4b = = 2m d = 7. 5s = 30 b = m = = t = u - 33 s = t = u = f = 18 f = = z 8 z = g = 17 g = = 98 - v v = 14. j - 38 = 36 j = q = 65 q = = n + 8 n = = 48 e e = c = 2 c = h = 9 h = = k - 42 k = 21. 9n = 36 n = = 12 + p p = 23. d + 65 = 81 d = = m 7 m =

90 STUDENT MANUAL MATH LESSON 16-1 Lesson 16 Understanding Inequalities You have learned how to write and solve basic equations, understanding that there are many different words and phrases that represent the various operations, and that you can often use mental math to solve equations. In this Lesson, you will learn how to determine the solution of inequalities. An inequality is a mathematical sentence that contains a or sign. < is the less than sign > is the greater than sign is the less than or equal to sign is the greater than or equal to sign is a solution to an inequality or not. To do so, replace the variable in the be more than one solution to an inequality. Example: Which of the numbers 2, 3, or 4 is a solution to the inequality X + 5 > 7?

91 STUDENT MANUAL MATH LESSON 16-2 Example: Which of the numbers 5, 6, or 7 is a solution to the Example: Which of the numbers 10, 11, or 12 is a solution to the solution to a basic inequality for some problems, more than one answer may be correct. 1. Which of the numbers 7, 8, or 9 is the solution to the inequality X + 3 > 11? 2. Which of the numbers 1, 2, or 3 is the solution to the inequality 3. Which of the numbers 20, 21, or 22 is the solution to the inequality X 9 > 12? Lesson Wrap-Up: Complete the Lesson 16 Practice Sheet.

92 STUDENT MANUAL MATH LESSON 16-3 Lesson 16 Practice Sheet Determine if the inequality is a true or false statement < 12x 2. a > 7 > 10 w x = b < 28 a = < 15j w = > 27 - q 7. c 4 b = 6 = 9 c = 36 j = p < 61 p = 12 q = t t = 4 > > 7u d > < 8n v = > 7u u = h d = 12 5 < 23 h = 115 n = > 12 f f = a < j < > 28 + p a = > 9r j = s > 29 p = > 30 - q r = 9 s = 11 q = 27

93 STUDENT MANUAL MATH LESSON 17-1 Lesson 17 The Algebraic roperties P Introduction to Algebra Assessment Commutative Property This property explains that if the order of the addends or factors in an expression or equation is changed, the sum or product stays the same. Example: = both equal 9 Example: 7 2 = 2 7 both equal 14 Associative Property This property explains that however you group the addends or factors, it will not change the sum or product. Example: ( ) + 4 = 8 + ( ).. both equal 17 Example: ( 6 x 6 ) x 2 = 6 x ( 6 x 2 )..both equal 72 Distributive Property This property explains that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. Example: 4 x ( ) = ( 4 x 7 ) + ( 4 x 3 ) = 40 Identity Property of addition This property states that the sum of zero and any number is that number. Example: = 7 Identity Property of multiplication This property states that the product of one and any number is that number. Example: 7 x 1 = 7

94 STUDENT MANUAL MATH LESSON 17-2 Identify the property that is represented in each Example: x ( ) = ( 13 x 5 ) + ( 13 x 2 ) = = ( 5 x 6 ) x 3 = 5 x ( 6 x 3) 5. 3 x 1 = 3 Lesson Wrap-Up: name which one it is.

95 STUDENT MANUAL MATH LESSON 17-3 Introduction to Algebra Assessment Part 1: Find the value of each expression r for r = j for j = a + a for a = c for c = s 4 for s = d + 4 for d = 5 Part 2: Write the expression or equation represented by the word problem. 7. quotient of 12 and a number r less a number t 9. a number p increased by add 5 to a number a 11. take away a number d from multiplied by a number b

96 STUDENT MANUAL MATH LESSON 17-4 Part 3: Solve each equation using mental math d = n + 54 = = k = z r = = 67 - t Part 4: Tell whether each statement is true or false f < 20 f = b > 24 3 b = 36 c = 16

97 STUDENT MANUAL MATH LESSON 18-1 Lesson 18 Finding atterns P In this Lesson, you will learn how to estimate the product of various types of problems using multiples of 10. A multiple of a whole number is all of the whole numbers that are products of the given number and another whole number. You should know at least the multiples of 10 up to 100, as you have learned how to count by 10 (10, 20, 30, 40, 50, 60, 70, 80, 90, 100). If you use a multiple of 10 to estimate the answer to a multiplication problem, you count the number of zeros in the rounded numbers and put that number at the end of the product of the non-zero terms. Begin this exercise by estimating the products of 2-digit numbers: Directions: If the ones digit in the number is 0-4, round down. For Example, round 34 to 30 because the ones digit is a 4. If the ones digit in the number is 5-9, round up. For Example, round 58 to 60 because the ones digit is an 8. Round each number in the multiplication problem. Finally, multiply the two numbers together to get the estimation. Example: Estimate 45 x 12 Example: Estimate 78 x 33

98 STUDENT MANUAL MATH LESSON 18-2 Example: Estimate 42 x 97 Now you will learn how to multiply 3-digit numbers: If the tens digit is 0-4, round down. If the tens digit is 5-9, round up. Round each number in the multiplication problem. Then, multiply the two numbers together to get the estimate. Example: Estimate 123 x 76 Example: Estimate 487 x 43 Example: Estimate 213 x 69 Lesson Wrap-Up: Complete the Lesson 18 Practice Sheet.

99 STUDENT MANUAL MATH LESSON 18-3 Lesson 18 Practice Sheet Estimate each product by rounding x x x x x x x x x x x x x x x x x x x x x 639

100 STUDENT MANUAL MATH LESSON 19-1 Lesson 19 Multiplication of 1-Digit Numbers In this Lesson, you will learn how to multiply numbers of various digit amounts by a onedigit number. The ability to multiply correctly is an essential skill for real-life. These are the steps involved in multiplication by a one-digit number: Step 1: necessary. Step 2: necessary. Step 3: necessary. You can continue this process through the thousands place, and through even greater places. Keep multiplying one digit at a time from right to left until all numbers have been multiplied. Complete the following Examples illustrating the steps involved in multiplication by a one-digit number. Examples: 145 x 6 5,673 x x 2

101 STUDENT MANUAL MATH LESSON 19-2 Complete the following Practice Problems. Remember that you can check your work by estimating as you learned in Lesson x 3 = 2. 3,420 x 4 = x 6 = 4. 1,329 x 5 = Lesson Wrap-Up: Complete the Lesson 19 Practice Sheet.

102 STUDENT MANUAL MATH LESSON 19-3 Lesson 19 Practice Sheet Multiply. Check your answer by estimating x x x x x x x x x x x x x x x x ,284 x ,972 x ,870 x ,663 x ,567 x ,495 x ,647 x ,038 x 3

103 STUDENT MANUAL MATH LESSON 20-1 Lesson 20 Multiplication by a 2-Digit Number Multiplication Assessment In Lesson 19, you learned how to multiply a number by a one-digit value. In this Lesson, you will add two-digit values to the list and work through multiplication by a two-digit number. These are the steps involved in multiplication by a two-digit number: Step 1: digit value. Step 2: digit value. Step 3: Add the two products together. the ones place, just as if you were multiplying by any other multiple of 10. Complete the following Examples. Remember that you can check your answers using estimation. Examples: 1, x 34 x 12 x 20

104 STUDENT MANUAL MATH LESSON 20-2 Complete the following Examples to practice multiplication: x 32 = 2. 9,873 x 12 = x 34 = 4. 2,356 x 19 = Lesson Wrap-Up: Complete the Lesson 20 Practice Sheet.

105 STUDENT MANUAL MATH LESSON 20-3 Lesson 20 Practice Sheet Multiply. Check your answer by estimating x x x x x x x x x x x x x x x x ,582 x ,656 x ,019 x ,135 x ,466 x ,141 x ,737 x ,681 x 45

106 STUDENT MANUAL MATH LESSON 20-4 Multiplication Assessment First, estimate each product. Then, multiply to solve the problem x x x x x x x x x x x x x x x x x ,362 x x x x ,245 x x x 3

107 STUDENT MANUAL MATH LESSON 21-1 Lesson 21 Patterns with Decimal Multiplication This Lesson begins a series of Lessons on Decimal Multiplication. When multiplying decimals, the steps are very similar to multiplication of whole numbers, but the decimal point in each factor must be taken into account. Think about what would happen to the size of a pencil if you were to multiply the length by 10. Obviously the length of the pencil would be ten times greater. When working with decimals and multiplying by multiples of 10 there will be a pattern in the decimal points. Every time you multiply by a multiple of 10 (10, 100, 1000, etc), the decimal point moves one place to the right for each multiple by which you are multiplying. Here are several Examples of these types of problems that use this pattern. The pattern is helpful to remember to quickly solve decimal multiplication problems in which you are multiplying by a multiple of 10. Example: 0.56 x 1 = 0.56 The decimal point does not move x 10 = 5.6 The decimal point moves one place to the right x 100 = 56. The decimal point moves two places to the right x 1000 = 560. The decimal point moves three places to the right. The number of zeroes that are in the multiple of 10 is the number of places you move the decimal point to the right in the answer. Example: x 1 = x 10 = x 100 = x 1000 = 3.2

108 STUDENT MANUAL MATH LESSON 21-2 Now solve the following problems and prepare to discuss the answers: x 100 = x 10 = x 1000 = x 100 = x 10 = Lesson Wrap-Up: Complete the Lesson 21 Practice Sheet.

109 STUDENT MANUAL MATH LESSON 21-3 Lesson 21 Practice Sheet Multiply x x x x x x x x x x x x x x x x x x x x 100

110 STUDENT MANUAL MATH LESSON 22-1 Lesson 22 Multiplying with Decimals Multiplying decimals is very similar to multiplication of whole numbers. Follow these steps: Step 1: Multiply as you would with a whole number. When setting up the problem, make sure to align the decimal points. Step 2: Count the number of decimal places in the factors that you used to multiply. Step 3: Begin at the right side of your answer and count over the number of places you found in step 2. Put the decimal point in this place. Begin with these decimal multiplication Examples: Example: Multiply 1.2 x 3.5 Other Examples: 1. Multiply x Multiply 1.99 x Multiply x 3.4 Lesson Wrap-Up: Complete the Lesson 22 Practice Sheet.

111 STUDENT MANUAL MATH LESSON 22-2 Lesson 22 Practice Sheet Multiply x x x x x x x x x x 5.02

112 STUDENT MANUAL MATH LESSON 23-1 Lesson 23 More Multiplication with Decimals In Lesson 22, you learned how to multiply decimals. Practice what you ve learned by doing the following Examples: x 3.4 = x 12.3 = x 4.5 = In real life, you might also be asked to solve word problems involving decimal multiplication. Can you think of an instance in real-life when you may need to multiply decimals? Here are some ways to solve word problems involving decimal multiplication: Sam buys 6 candy bars at $1.10 each. How much does he spend? Mark works 40 hours per week at $12.89 per hour. How much does he make in a week? How much does he make in 4 weeks?

113 STUDENT MANUAL MATH LESSON 23-2 Amy wants to increase the size of her closet wall by 2 times. If the wall currently is 3.2 feet, how large will the new closet wall be? Lesson Wrap-Up: Complete the Lesson 23 Practice Sheet. Multiplying money is just one use of decimal multiplication.

114 STUDENT MANUAL MATH LESSON 23-3 Lesson 23 Practice Sheet Solve each problem. school supplies. She bought a notebook, two pocket folders, seventeen pencils, three erasers, a pair of scissors, a box of colored pencils, two black pens, eight blue pens, one ruler, and four packages of paper. Paper $1.29/package Rulers $0.99 each Pens $0.73 each Colored Pencils $3.43/box Scissors $4.42/pair Erasers $0.15 each Pencils $0.16 each Pocket Folders $0.32 each Notebook $4.41 each Total Spent: 2. It was my fault that we arrived at the musical later than we had expected. I had to stop to get gas. I put seven gallons of gas in my car. Each gallon cost me $1.76. How much did I spend on gas? Total Spent:

115 STUDENT MANUAL MATH LESSON Jonathan s brother Austin is entering Mountain Community College this year. When he registered, he was told he could pay for his meals in cash, or he could buy a meal ticket for the whole year. If he pays cash, each meal he eats will cost $5.62. A meal ticket for the year pays for two meals per day for one hundred sixty-eight days and costs $1, If Austin eats 2 meals per day for one hundred sixty-eight days, how much would he save if he bought the meal ticket instead of paying for his meals in cash? Total Saved: 4. Jason is a dairy farmer, and his cow gives 10.9 gallons of milk a day. How many gallons does she give in one hundred twenty-one days? Total Gallons: 5. Morgan was hungry. She looked in the pantry but found nothing fun to eat. She looked under her bed where she often kept cookies but only found one empty cookie box. She would have to buy something. She took $2.64 out of her hiding place and walked to the store. She wanted as many forty-four cent candy bars as she could get. How many candy bars can she buy? Total Bars:

116 STUDENT MANUAL MATH LESSON 24-1 Lesson 24 Understanding Zeros Multiplying with Decimals Assessment When multiplying decimals, it is a common mistake to place the decimal allow you to practice multiplying decimals that have zeros in the answer so you can practice avoiding this mistake. Remember the steps involved with multiplication of decimals: Step 1: Multiply as you would with whole numbers. When setting up the problem, make sure to align the decimal points. Step 2: Count the number of decimal places in the factors that you used to multiply. Step 3: Begin at the right side of your answer and count over the number of places you found in step 2. Put the decimal point in this place. It is important to remember that you must put the placement zero in the second row of your answer to a multiplication problem (when needed); otherwise, the solution will be incorrect. Work through the following Examples to practice decimal multiplication Examples: x x x. 2 Lesson Wrap-Up: Complete the Lesson 24 Practice Sheet.

117 STUDENT MANUAL MATH LESSON 24-2 Lesson 24 Practice Sheet Multiply x x x x x x x x x x x x x x x x 0.6

118 STUDENT MANUAL MATH LESSON 24-3 Multiplying with Decimals Assessment Solve each decimal multiplication problem x x x x x x x x x x x x x x x x x x x x x x x x 0.924

119 STUDENT MANUAL MATH LESSON 25-1 Lesson 25 Estimating otients Qu In this Lesson, you will begin your study of Division. You will learn how to estimate the answer (quotient) of a division problem using compatible numbers, which are numbers that can be easily computed mentally. Compatible numbers are most commonly the multiples of a particular number In order to make an estimate on a division problem, look at the following Examples to determine the steps you will work through: Example: 557 / 8 Step 1: Determine the multiples of 8. Step 2: Determine which of the multiples above is closest to 55. Step 3: Create your two compatible numbers 8 and 560, instead of 557. Step 4: Divide 560 / 8 = 70. Step 5: Estimate that 557 / 8 is about 70.

120 STUDENT MANUAL MATH LESSON 25-2 Example: 23,120 / 4 Step 1: Determine the multiples of 4. Step 2: Determine which of the multiples above is closest to 23. Step 3: Create your two compatible numbers 4 and 24,000. Step 4: Divide 24,000 / 4 = 6,000 Step 5: Estimate that 23,120 / 4 is about 6,000. Here are a few problems to practice with to estimate the quotients: / ,672 / / 5 Lesson Wrap-Up: Create 10 problems to trade with a fellow student or your teacher to work to practice estimating the quotient.

121 STUDENT MANUAL MATH LESSON 26-1 Lesson 26 Dividing by 1-Digit Divisors This Lesson will introduce you to the steps involved in performing a long division problem with a 1-digit divisor. Before you begin, here are the three main parts of a division problem: Dividend The dividend is the number that is to be divided in a division problem. Divisor The divisor is the number that divides the dividend in a division problem. Quotient The quotient is the solution to a division problem. Remainder The value left over when the divisor does not go evenly into the dividend. Here are the steps involved in long division: Step 1: Look at the number in the hundreds place in the dividend. If the divisor is less than this number, divide this number by the divisor. If the divisor is greater than this number, move to the tens place (or ones place if necessary). Step 2: Multiply the new quotient by the divisor and place this number underneath the dividend. Step 3: Subtract this product by the dividend above it. Step 4: If there are remaining numbers in the dividend, bring the next number down and continue this process.

122 STUDENT MANUAL MATH LESSON 26-2 Let s look at several Examples: Example: 382 / 4 Example: 127 / 5 Example: 589/ Practice dividing the following problems on your own: / / /6 Lesson Wrap-Up: Complete the Lesson 26 Practice Sheet.

123 STUDENT MANUAL MATH LESSON 26-3 Lesson 26 Practice Sheet Divide

124 STUDENT MANUAL MATH LESSON 27-1 Lesson 27 Using Zeros in Division Sometimes when you are completing division problems you will have a zero in the quotient. It is important to remember that the zero cannot be removed from the quotient of the problem. Let s look at some Example Problems where this occurs: Whenever you cannot divide your divisor evenly into the next number in the dividend, you will write 0 in the quotient, and carry the number in the dividend down. digit number that you can divide by the divisor. Example: 422/ Example: 926/ Lesson Wrap-Up: Complete the Lesson 27 Practice Sheet.

125 STUDENT MANUAL MATH LESSON 27-2 Lesson 27 Practice Sheet Divide and show the remainder, if there is one. 1. 8, , , , , , , , , , , ,

126 STUDENT MANUAL MATH LESSON 28-1 Lesson 28 Dividing by 2-Digit Divisors Review what you learned in Lesson 27 by completing the following Practice Problems: / / /6 The steps involved in dividing when you have a 2-digit divisor are the same as with a 1-digit divisor; however, you use the entire 2-digit number when dividing. Let s review those steps: Step 1: Look at the number in the hundreds place in the dividend. If the divisor is less than this number, divide this number by the divisor. If the divisor is greater than this number, move to the tens place (or ones place if necessary). Step 2: Multiply the new quotient by the divisor and place this number underneath the dividend. Step 3: Subtract this product from the dividend above it. Step 4: If there are remaining numbers in the dividend, bring the next number down and continue this process.

127 STUDENT MANUAL MATH LESSON 28-2 Use these Examples to practice dividing with a 2-digit divisor: Example: 34/ Example: 345/ Example: 762/ Lesson Wrap-Up: Complete the Lesson 28 Practice.

128 STUDENT MANUAL MATH LESSON 28-3 Lesson 28 Practice Sheet Divide, and round answers off to the nearest ten thousandths (if you had an answer of , you should round off to ) , , , , , , , , , ,089 37

129 STUDENT MANUAL MATH LESSON , Part 2: Order the decimals from least to greatest , 0.32, 0.12, , 0.587, 0.593, , 0.131, 0.33, 0.3

130 STUDENT MANUAL MATH LESSON 28-6 Decimals Assessment Part 1: For each decimal given, write the decimal in expanded form and in word form Expanded: Word: Expanded: Word: Expanded: Word: 4. 3,126.9 Expanded: Word: Expanded: Word: Expanded: Word:

131 STUDENT MANUAL MATH LESSON 28-7 Part 2: For each pair of decimals, determine whether the two decimals are equivalent or not equivalent and and and and and and 3.24 Part 3: Compare each pair of decimals using <, >, or =

132 STUDENT MANUAL MATH LESSON 28-8 Part 4: Order each set of decimals from least to greatest , 0.12, 0.36, 0.41, , 12.03, 12.75, 12.12, 12.34, , , , , , PRINT

133 STUDENT MANUAL MATH LESSON 29-1 Lesson 29 More Practice with Division Division Assessment For the last 4 Lessons, you have been studying division if you have any questions about division, you should ask your teacher during this class. Remember the basic steps for division: Step 1: Look at the number in the hundreds place in the dividend. If the divisor is less than this number, divide this number by the divisor. If the divisor is greater than this number, move to the tens place (or ones place in necessary). Step 2: Multiply the new quotient by the divisor and place this number underneath the dividend. Step 3: Subtract this product from the dividend above it. Step 4: If there are remaining numbers in the dividend, bring the next number down and continue this process. For review, complete the More Practice with Division Practice Sheet. When you are done, you will complete the Division Assessment. Lesson Wrap-Up: Complete the Division Assessment, and go over it with your teacher when you are done. Be sure to ask any questions you may have!

134 STUDENT MANUAL MATH LESSON 29-2 More Practice with Division Divide, and round answers off to the nearest ten thousandths (if you had an answer of , you should round off to ) , , , , , , , , , , , ,277 53

135 STUDENT MANUAL MATH LESSON 29-3 Division Assessment Use a separate sheet of paper to divide each problem correctly. Round answers off to the nearest ten thousandths (if you had an answer of , you should round off to ) , , , , , , , ,

136 STUDENT MANUAL MATH LESSON 30-1 Lesson 30 Using Patterns When Dividing Decimals Explain how you would answer the following question: If Sally is trying to divide $4000 equally into 6 envelopes, how much money should she put into each one? How much should she put into 6 envelopes if she is trying to divide $400? How about $40? When dividing, it is often helpful to look for a pattern and use mental math as much as possible. For this problem, you can see a pattern in the answers: 4,000 / 6 = / 6 = / 6 = 6.67 Remember that if the divisor in the problem is greater than the dividend of the problem, the quotient will be a decimal less than one. Use patterns to mentally solve division problems because every time that there is one less zero in the dividend of a problem, the decimal point in the quotient moves one place to the left. Use a pattern to solve these division problems: Example: Divide 3 / 5 Also 30 / 5 = 6. Therefore, following the pattern, you can deduce that 3 / 5 = 0.6, by moving the decimal point one place to the left.

137 STUDENT MANUAL MATH LESSON 30-2 Example: Divide 4.3 / 2 Then you can realize that 43 / 2 = Finally, by understanding the pattern, you should be able to see that 4.3 / 2 = Example: Divide 9.7 / 6 Therefore by seeing a pattern, 97 / 6 = Finally, using this pattern, see that 9.7 / 6 = Complete the following Practice Problems in order to practice using patterns to divide decimals: / / / / 3 Lesson Wrap-Up: Complete the Lesson 30 Practice Sheet.

138 STUDENT MANUAL MATH LESSON 30-3 Lesson 30 Practice Sheet Directions: Solve the following problems by dividing. As you work, look for patterns which help you solve the problem more quickly

139 STUDENT MANUAL MATH LESSON 31-1 Lesson 31 Dividing Decimals by Whole Numbers In this Lesson, you will divide decimals by whole numbers. This is almost no different than dividing any other 2, 3, or 4-digit number except that you must keep the decimal in place. The decimal moves straight up on the division symbol in exactly the same way as it moves down in addition and subtraction! When completing these problems, remember the following steps: Step 1: Write the problem as a normal division problem. Step 2: Begin by dividing the whole numbers. Step 3: Put the decimal point in the answer directly above where the decimal point appears in the problem. Step 4: Continue solving the division problem as you have learned. Complete these Examples to practice dividing decimals by whole numbers. Example: Did you see how the decimal moved straight up? Try it on some more Practice Problems

140 STUDENT MANUAL MATH LESSON 31-2 How would you solve a word problem involving division? 1. Jannie s friends are all going together to purchase her a birthday present. There are 6 friends, and they decide to buy a new DVD player that costs $ How much will each friend need to give to cover the cost? If Susan works for 6 hours and makes $ total, what is her pay per hour? Lesson Wrap-Up: Complete the Lesson 31 Practice Sheet.

141 STUDENT MANUAL MATH LESSON 31-3 Lesson 31 Practice Sheet Divide = = = = = = = = = = = =

142 STUDENT MANUAL MATH LESSON 32-1 Lesson 32 Fraction onversions C In this Lesson, you will learn to convert fractions to decimals by using division. Before you do this, take a moment to review comparing fractions by completing the following warm-up exercises: Compare the following decimals using <, >, or =: ½ ¾ When being asked to compare fractions, it is helpful to know what the decimal equivalent of the fraction is. To determine these decimal equivalents, you can use decimal division. The steps to convert a fraction to a decimal are simple: Step 1: Write the fraction as a division problem dividing the numerator by the denominator. Step 2: Divide as you would if you were dividing a decimal by a whole number. Put the decimal point in the quotient directly above where it occurs in the dividend. Step 3:

143 STUDENT MANUAL MATH LESSON 32-2 Examine the following Examples: Convert the following fractions to decimals: Complete these Practice Problems: 1. Convert 8 11 to a decimal 2. Convert 7 9 to a decimal 3. Convert to a decimal Lesson Wrap-Up: Complete the Lesson 32 Practice Sheet.

144 STUDENT MANUAL MATH LESSON 32-3 Lesson 32 Practice Sheet Convert each fraction to a decimal - round answers off to the nearest thousandth if needed ½ ¼

145 STUDENT MANUAL MATH LESSON 33-1 Lesson 33 Decimal onversions C In Lesson 32, you learned how to convert a fraction to a decimal. Practice this skill with the following problems before we begin Lesson 33: 1. Convert 6 15 to a decimal 2. Convert 8 9 to a decimal 3. Convert 9 11 to a decimal This Lesson will cover making conversions from a decimal to a fraction. A necessary understanding of place value will be important when making these conversions. The 1 is in the tens place. The 0 is in the ones place. The 2 is in the tenths place. The 3 is in the hundredths place. The 4 is in the thousandths place.

146 STUDENT MANUAL MATH LESSON 33-2 In order to convert a decimal to a fraction, you will follow these steps: Step 1: Determine the place value of the last digit in the number. Step 2: This place value will become the denominator of your fraction. Ones place denominator will be 1 Tenths place denominator will be 10 Hundredths place denominator will be 100 Thousandths place denominator will be 1000 Step 3: Take the decimal point out of the decimal and place this number in the numerator of the fraction. Step 4: Reduce this fraction if possible. Now look at these Examples to understand how to make these conversions: Convert 1.23 to a fraction. Step 1: The place value of the last digit is the hundredths place. Step 2: Therefore, the denominator of the fraction will be 100. Step 3: Taking the decimal point out, we have a numerator of 123. Step 4: The fraction is Convert 5.6 to a fraction. Step 1: The place value of the last digit is the tenths place. Step 2: Therefore, the denominator will be 10. Step 3: Taking the decimal point out, we have a numerator of 56. Step 4: The fraction is This fraction can be reduced because both the numerator and denominator are even. Dividing both portions of the fraction by 2 gives a new fraction of Convert to a fraction Step 1: The place value of the last digit is the thousandths place. Step 2: Therefore, the denominator of the fraction is Step 3: Taking the decimal point out, we have a numerator of Step 4: The fraction is

147 STUDENT MANUAL MATH LESSON 33-3 Practice conversions by completing the following problems: 1. Convert 8.97 to a fraction 2. Convert 4.2 to a fraction 3. Convert 6 to a fraction 4. Convert to a fraction Lesson Wrap-Up: Complete the Lesson 33 Practice Sheet.

148 STUDENT MANUAL MATH LESSON 33-4 Lesson 33 Practice Sheet Convert each decimal to a fraction, and reduce when possible. 1. one tenth thousandths two tenths tenths 10. eighty hundredths

149 STUDENT MANUAL MATH LESSON tenths 14. ten and one tenth ninety-three hundredths thousandths sixty-nine hundredths tenths 22. two and four hundred twentytwo thousandths two and two hundred forty-four thousandths

150 STUDENT MANUAL MATH LESSON 34-1 Lesson 34 Dividing Decimals by Decimals Decimal ivision D Assessment In Lesson 33, you practiced converting decimals to Fractions. In this Lesson, you will learn how to divide decimals by other decimals. Warm up by answering these division problems: / / / / 7 Let us discuss the answers to the Practice Problems. How would you solve the following problem? Mark has been putting $0.35 into his piggy bank every day. He currently has $9.10 saved. How long has he been putting money into his piggy bank?

151 STUDENT MANUAL MATH LESSON 34-2 What steps are involved in solving a problem where you need to divide a decimal by a decimal? Step 1: Make the divisor a whole number by multiplying the divisor and the dividend by 10, 100, 1000 etc. Step 2: Place the decimal point in the quotient directly above the decimal point in the dividend. Step 3: Divide as you would with whole numbers. Practice the division of a decimal by a decimal with the following Examples: Example: Step 1 Multiply 2.3 x 10 = 23 Multiply 8.5 x 10 = 85 Step 2 Place the decimal point at the end of the dividend. Step

152 STUDENT MANUAL MATH LESSON 34-3 Practice solving division problems with more Examples: / / / / 0.18 Go through the following Examples of word problems and discuss them with the teacher: 1. Jannie paid $1.56 for 2.6 pounds of apples. How much did she pay per pound? $ per pound 2. which is 15.5 gallons, how much was the price per gallon of gas? $ per gallon Lesson Wrap-Up: Practice dividing decimals by decimals by creating three Practice Problems, trading them with a partner, and then having the partner complete the problems. Trade papers back and check each other s work. Complete the Lesson 34 Practice Sheet.

153 STUDENT MANUAL MATH LESSON 34-4 Lesson 34 Practice Sheet Part 1: Divide. Round answers to the nearest hundredth = 5. 6.

154 STUDENT MANUAL MATH LESSON 34-5 Part 2: Solve each word problem. 1. Marcus has $24.50 in quarters. How many quarters does Marcus have? quarters 2. Janice has a piece of fabric that is 9 feet wide. In order to complete her class project, she needs 25 pieces of fabric that are equal widths. What width should she make the pieces of fabric? feet 3. Peter saves $5.25 each week because he wants to buy a video game that is $ He has already saved $ How many more weeks does he need to continue saving in order to buy the video game? weeks

155 STUDENT MANUAL MATH LESSON 34-6 Decimal Division Assessment Part 1: Divide. Round answers off to the nearest thousandth if needed

156 STUDENT MANUAL MATH LESSON 34-7 Part 2: Convert each fraction to a decimal or decimal to a fraction Thousandths Two and seven tenths Three and forty-nine hundredths 25. Eight tenths thousandths 29. Five and three hundredths thousandths

157 STUDENT MANUAL MATH LESSON 35-2 Lesson 35 Working with Expressions In this Lesson, you will review what you have learned about algebraic expresions. Review the following expressions: Expression A mathematical phrase that includes a combination of numbers, variables and operation symbols. Variable A letter or a symbol used to represent one or more numbers. Evaluate Review the sample algebraic expression you learned in lesson 13: the second half. To represent the total points the team scored, you can use If you did not know how many points the football team scored in the second half, you could use a variable to represent the points. You can use any letter of the alphabet as a variable, although it is common to use a letter that corresponds to the piece of information it will represent. For example, to represent the total points scored in the second half of the football game we will use P. recall that the algebraic expression that would represent the total points scored in the game would be 21 + P.

158 STUDENT MANUAL MATH LESSON 35-3 Review the following key vocabulary: Addition Subtraction Multiplication Division Sum Difference Product Quotient Increase Decrease Factors Equally divided More than Less than Twice Divided by Plus Minus Times Greater than Fewer than Multiplied by

159 STUDENT MANUAL MATH LESSON 35-5 Lesson 35 Practice Sheet Part 1: Write each algebraic expression. Use your digital tools. 1. a number s minus divided by a number j a number n decreased by quotient of 12 and a number r less a number t 9. a number p increased by add 5 to a number a 11. take away a number d from multiplied by a number b more than a number m divided into a number x 15. a number u less than

160 STUDENT MANUAL MATH LESSON 35-6 Part 2: Evaluate each expression for the given variables. 21. (19 - r) + r for r = j for j = c for c = x for x = r + 8r + 3 for r = z - z for z = q - 88 for q = h + 18 for h = 56

161 STUDENT MANUAL MATH LESSON 36-1 Lesson 36 Working with Equations Review Lesson 35 by writing the following algebraic expressions: 1. A number decreased from more than a number X times the quantity of a number added to The quotient of X and Y. In this Lesson, you will use the skills you learned in Lesson 35 to write equations. An equation is a mathematical sentence that shows that two quantities are equal. The difference between an equation and an expression is that an equation includes an equal sign and an expression does not. Writing equations is a skill that is used in everyday life. Give an Example of when you might have to compose an equation in everyday life to solve a particular problem.

162 STUDENT MANUAL MATH LESSON 36-2 Working with equations: Part 1: Determine which of the following numbers is a solution to each equation by using mental math. Simply plug each number given into the equation for the given variable to determine if what number works to make the equation true. Example: 12X = , 12, 13 Example: 14 Y = 7 1, 2, 3 Example: 63 / A = 7 7, 8, 9

163 STUDENT MANUAL MATH LESSON 36-3 Part 2: Solve these word problems: 1. The concession stands at the football stadium sold 13,456 hot dogs during the 3 hour football game. If the same amount was sold each hour, how many hot dogs were sold every hour? Hot Dogs 2. The teacher has 3 dozen cookies to share among 10 students in the classroom. How many cookies should each student get if each student is to receive an equal amount? Cookies 25 books in a box, how many boxes does she need? Boxes Lesson Wrap-Up: Complete the Lesson 36 Practice Sheet.

164 STUDENT MANUAL MATH LESSON 36-4 Lesson 36 Practice Sheet Solve each problem. 1. There are 10 towels in the closet. If you put them into 2 equal groups, how many towels will be in each group? towels 2. groups, how many tiles will be in each group? tiles There are 48 cherries in the bucket. If you put them into 6 equal groups, how many cherries will there be in each group? cherries

165 STUDENT MANUAL MATH LESSON Grace put 21 stickers in her scrapbook. If she put them into 3 equal groups, how many stickers will be in each group? stickers 7. Kevin made 14 birthday cards. If he makes 2 equal groups, how many cards will there be in each group? cards 8. There are 18 stray dogs at the animal shelter. If they make 9 equal groups, how many dogs will be in each group? dogs

166 STUDENT MANUAL MATH LESSON somewhere in her room there are 6 spiders. Each spider has 8 legs. How many spider legs are in Sarah s room? legs 10. Alexis wanted to win the game. Each team member scored 9 points. There were 4 people on the team. How many points did the team score in all? points Alyssa sold 6 rabbits for 5 dollars each. How much money did Alyssa earn by selling her rabbits? $

167 STUDENT MANUAL MATH LESSON My brother has 5 friends. Each friend has 2 pets. How many pets do his friends have in all? pets 14. Anna is helping her mother get ready for dinner. Her mother asked her to put 8 cherry tomatoes on each salad. If there are 7 salads, how many cherry tomatoes will she need? cherry tomatoes 15. city workers. There were 6 people from each of 3 departments within the city. How many people were in the hallway? people 16. We heard the frog croak in the bushy shrubs. Then it stopped and became quiet. We estimated that it croaked 4 times every ten seconds. How many times did it croak in one minute? times

168 STUDENT MANUAL MATH LESSON 37-1 Lesson 37 Order of Operations Examine the following word problem: Anna is saving money to buy a new DVD player. She earned $5 during her plan, she earned $3 for every day that she made her bed. She correctly made her bed 6 days during that week. How much does Anna have saved thus far? Try to create an expression based on this word problem and then solve the expression. Whenever you are asked to solve an expression, which has more than one operation (addition, subtraction, multiplication, division, or parentheses), you must follow the Order of Operations, which is the set of rules used to solve a problem with multiple operations. The steps in the order of operations are as follows: Step 1: Complete anything inside of parentheses. Step 2: Solve any exponential expressions in the problem. You have not the operations included in steps 1, 3, and 4. Step 3: Multiply and divide from left to right in the problem. Step 4: Add and subtract from left to right in the problem.

169 STUDENT MANUAL MATH LESSON 37-2 Look at the following problems to help further explain the Order of Operations: (11 4) = 2. (6 x 9) 2 = (8 + 1) = 4. (15 3) + 5 x 2 = 5. 8 x x 8 =

170 STUDENT MANUAL MATH LESSON 37-3 Do the following Practice Problems and check with a partner to determine if you are correctly following the order of operations: (5 x 6) = (20 10) 1 = (50 10) + 3 = Lesson Wrap-Up: Complete the Lesson 37 Practice Sheet.

171 STUDENT MANUAL MATH LESSON 37-4 Lesson 37 Practice Sheet Solve each problem using the order of operations (54 2) (27 3) 3. ( ) (7 + 5) (33 3) (14 2-1) 9. ( ) (27 3) (5-1)

172 STUDENT MANUAL MATH LESSON 38-1 Lesson 38 Finding atterns P You will often use patterns to solve problems, understand designs, and determine the costs of things in everyday life. Examine this real-life Example: The teachers at Mason High School use a phone-calling tree to contact each other when school is canceled due to inclement weather. The tree is set up in the following way: The principal calls three teachers, each of those teachers calls three teachers, each of those 9 teachers calls three teachers, and so on. Example 1: Can you see the pattern in the above Example? The pattern is 1, 3, 9, 27. What would the next number in the pattern be? The answer is 81. Use the pattern in this Example to answer the following question: How many calling rounds will it take to call all 200 staff members in the school? what the rule is that is creating the pattern.

173 STUDENT MANUAL MATH LESSON 38-2 Study the following Examples and try to determine the next number in the pattern. Example 2: 3, 7, 11, 15, 19, 23, Example 3: 8, 40, 20, 100, Example 4: 15, 25, 45,, 115, 165

174 STUDENT MANUAL MATH LESSON 38-3 numbers: 1. 1, 5, 9, 13,, 2. 3, 6, 12, 24,, 3. 80, 40, 20,, , 60, 20, 24,, 12, 4 Lesson Wrap-Up: Complete the Lesson 38 Practice Sheet.

175 STUDENT MANUAL MATH LESSON 38-4 Lesson 38 Practice Sheet Find the missing number in each pattern. 1. 7, 28, 112, 448, 1,792, 7,168,, 114,688, 458, ,, 84, 77, 70, 63, 56, ,488, 784,, 16 4., 23, 29, 35, 41, 47, , 51, 306, 314, 1,884, 6. 30, 120,, 1,920, 7,680, 30,720, 122,880, 491, , 252, 243,, 72, 24, , 117, 105, 93, 81, 69, 57,, 33

176 STUDENT MANUAL MATH LESSON , 1,045, 209, 205, 41, , 110, 95, 80, 65, 50,, , 15, 45, 135,, 1,215, 3, , 88, 352, 1,408, 5,632, 22,528, 90,112, 360,448, ,, 73, 92, 111, 130, ,935, 6,187, 6,185,, 1,235, 247, 245, 49, , 104, 416, 1,664, 6,656,, 106,496, 425, ,, 9, 7, 5, 3

177 STUDENT MANUAL MATH LESSON 39-1 Lesson 39 Working with Functions A function is a relationship between two quantities in which one depends upon the other. Functions can be used to solve problems when there is more than one variable. You will be given a function rule, and be asked to complete the function table applying this rule to the input values. Look at the following Examples: Example: Rule: b = 9c Input (c) Output (b) Example: Rule: y = 3x + 1 Input (x) Output (y) Example: Rule: a = 10 + b Input (b) Output (a) Lesson Wrap-Up: Complete the Lesson 39 Practice Sheet.

178 STUDENT MANUAL MATH LESSON 39-2 Lesson 39 Practice Sheet Complete each function table. 1. Rule: n = 12 n Input d Output n 2. Rule: y = 5b Input b Output y 3. Rule: a = 77 - q Input q Output a 4. Rule: u = k - 35 Input k Output u 5. Rule: u = v x 8 Input v Output u 6. Rule: x = g - 7 Input g Output x

179 STUDENT MANUAL MATH LESSON Rule: e = d + 18 Input d Output e 8. Rule: j = q - 10 Input q Output j Rule: a = 60 b Input b 2 30 Output a Rule: k = s 8 Input s Output k Rule: y = z + 3 Input z Output y Rule: r = 5t Input t 2 12 Output r

180 STUDENT MANUAL MATH LESSON 40-1 Lesson 40 The Commutative and Associative Properties The commutative property states that you can multiply numbers in any order. For Example, 5 x 7 is the same thing as 7 x 5. Changing the order in a multiplication problem will not change the answer to the problem. The associative property states that if you are using grouping symbols in a multiplication problem, you can move the grouping symbols around and you will not change the answer to the problem. For example, 2 x (4 x 5) equals 40, and if you move the parentheses and rewrite the problem as (2 x 4) x 5, you will still get 40 as the product. Understanding the commutative and associative properties is a necessary skill because knowledge of these properties will help you learn how to solve equations correctly. What property is being used in the following Examples? 1. 4 x 5 = 5 x X = X(8) 3. (9 x 6) x 4 = 9 x (6 x 4) 4. (Y x 3) x 2 = Y x (3 x 2) Lesson Wrap-Up: Explain the difference, in writing, between the Commutative and Associative properties.

181 STUDENT MANUAL MATH LESSON 41-1 Lesson 41 The Identity Property In this Lesson, you will learn to use the identity property of multiplication that states that if you multiply any number by 1, the product will equal the original number; and, if you multiply any number by 0, the product will equal 0. This means that identity values are 1 and 0 for multiplication. For Example: 6 x 1 = 6 (the original factor), and 6 x 0 = 0. missing value for each variable: 1. 5 x 2 = Y x 5 2. (6 x 3) x 1 = Y x (3 x 1) 3. 8 x (4 x 2) = (8 x Y) x 2 Work through the following Examples and identify the missing value for the variable utilizing the identity property x 0 = Y 2. 8 x 1 = Y x Y = 13 Lesson Wrap-Up: Complete the Lesson 41 Practice Sheet.

182 STUDENT MANUAL MATH LESSON 41-2 Lesson 41 Practice Sheet Identify the correct property illustrated in each Example. Then, tell the missing value in each expression x 2 = 2 x 2. 8 x = x (6 x 5) = ( x 6) x 5 4. x 1 = x 3 = 3 x 6. (10 x 2) x = 10 x (2 x 5) x = x = 3 x x (4 x 8) = (16 ) x x 1 = 12

183 STUDENT MANUAL MATH LESSON 42-1 Lesson 42 The Distributive Property Division Algebra in Assessment You have learned three of the four algebraic properties and how they apply distributive property. The distributive property states that if you have a number in front of grouping symbols (parentheses), you should multiply through, or distribute, that number to each number inside the grouping symbols. The following are Examples of the distributive property: Example: 4 x (3 + 7) = Example: 8( ) = Example: 3 x ( Y + 2) for Y=9 = Sometimes you will see a multiplication sign in between the number in front of the parentheses and the parentheses, and sometimes the number will be directly next to the parentheses without a sign. In either case, you should multiply the number in front of the grouping symbols through to every number in the parentheses.

184 STUDENT MANUAL MATH LESSON 42-2 Complete these Practice Problems to work on and check for your understanding of the Distributive Property: 1. 4(6 + 7) = 2. 8( ) = 3. 6(10 + 9) = Lesson Wrap-Up: Complete the Lesson 42 Practice Sheet.

185 STUDENT MANUAL MATH LESSON 42-3 Lesson 42 Practice Sheet 1. 6 (80 + 1) 2. 5 (16 + 2) 3. 3 (8 + 5) 4. 8 x (8 + 6) 5. (2 50) + (2 7) 6. 9 (30 + 5) 7. 5 x (7 + 9) 8. 4 (9 + 7) 9. 3 (2 + 4) (10 + 7) 11. (9 60) + (9 6) (10 + 2) (60 + 3) (7 + 1) (4 + 9) 16. (2 80) + (2 6) (80 + 4) 18. (7 10) + (7 1) (2 + 3) (60 + 7) 21. (8 80) + (8 7) (90 + 5) 23. (4 60) + (4 1) (8 + 2)

186 STUDENT MANUAL MATH LESSON 42-4 Division in Algebra Assessment Part 1: Write each word problem as an algebraic expression. 1. a number u times 5 2. a number y less plus a number t 4. difference of 47 and a number k divided into a number x 6. a number w minus 14 Part 2: Solve each algebraic expression. 7. q 4 for q = t for t = h for h = k - 52 for k = a + 42 for a = u + 17 for u = b - 2 for b = c - 10 for c = y + 41 for y = 5

187 STUDENT MANUAL MATH LESSON 42-5 Part 3: Solve each problem using the order of operations Part 4: Solve each word problem. 25. A number plus 40 is plus a number is A number divided by 11 is The difference between 85 and a number is divided by a number is Twelve times a number is 96.

188 STUDENT MANUAL MATH LESSON 43-1 Lesson 43 Divisibility You can probably tell if numbers are even or odd just by looking to warm up, tell whether these numbers are odd or even , How do you know whether a number is even or odd? Telling if a number is divisible by two (even) is just one pattern of divisibility. Every number from 2 to 10 has its own pattern.

189 STUDENT MANUAL MATH LESSON 43-2 This table explains the divisibility rules: A Number is Divisible by Divisible Not Divisible 2 if the last digit is even (0,2,4,6,8) 10, if the sum of the digits is divisible by if the last two digits form a number divisible by if the last digit is a 0 or ,236 6 if the number is divisible by 2 and if the last three digits form a number divisible by 8 5,336 8,102 9 if the sum of the digits is divisible by if the last digit is Is the number 38 divisible by 2,3,4,5,6,8,9, or 10? Write yes or no next to the number. Use your digital tools

190 STUDENT MANUAL MATH LESSON 43-3 Here are some more Examples for you to try on your own: Determine whether each number is divisible by 2,3,4,5,6,8,9, or Lesson Wrap-Up: Complete the Lesson 43 practice sheet, and review it with your teacher when you are done.

191 STUDENT MANUAL MATH LESSON 43-4 Lesson 43 Practice Sheet Fill in the divisibility table for each number by writing yes or no if the number is divisible by the indicated value. 1. 1,512 by 4 by 5 by 7 by 8 by ,958 by 2 by 3 by 7 by 8 by by 2 by 3 by 7 by 9 by by 2 by 3 by 7 by 8 by ,617 by 6 by 7 by 8 by 9 by ,646 by 2 by 3 by 6 by 7 by 9

192 STUDENT MANUAL MATH LESSON by 2 by 3 by 5 by 8 by ,285 by 4 by 5 by 10 by 11 by by 2 by 5 by 7 by 9 by ,659 by 3 by 5 by 7 by 10 by by by 7 by 8 by 9 by 13 by 2 by 5 by 6 by 11 by 13

193 STUDENT MANUAL MATH LESSON 44-1 Lesson 44 Prime and Composite Numbers In Lesson 43 you worked with patterns of divisibility. Sometimes, a number is only divisible by itself and 1 other numbers are divisible by lots of composite. The difference between a prime and composite number is: Prime number A whole number greater than 1 with only 1 and itself as factors. Composite number A whole number greater than 1 that has more than 2 factors. The numbers 0 and 1 are neither prime nor composite. Why is this? The best way to determine whether a number is prime or composite is to Example: If you want to determine whether 36 is prime or 36 / 1 = 36 therefore 1 and 36 are factors of / 2 = 18 therefore, 2 and 18 are factors of / 3 = 12 therefore, 3 and 12 are factors of / 4 = 9 therefore, 4 and 9 are factors of / 6 = 6 therefore, 6 is a factor of 36. Since 36 contains the factors 1,2,3,4,6,9,12,18, and 36, and this is more than 2 factors, 36 is a composite number.

194 STUDENT MANUAL MATH LESSON 44-2 Complete the following Examples to determine whether the numbers are prime or composite: You can solve for prime numbers by division, but in time you will memorize most of them under 100. Have your teacher quiz you to see how many you already know! Lesson Wrap-Up: Complete the Lesson 44 Practice Sheet.

195 STUDENT MANUAL MATH LESSON 44-3 Lesson 44 Practice Sheet Classify each number as prime or composite Prime Prime Prime Prime Composite Composite Composite Composite Prime Composite Prime Prime Prime Composite Composite Composite Prime Prime Prime Prime Composite Composite Composite Composite Prime Prime Prime Prime Composite Composite Composite Composite Prime Prime Prime Prime Composite Composite Composite Composite Prime Prime Prime Prime Composite Composite Composite Composite

196 STUDENT MANUAL MATH LESSON 45-1 Lesson 45 Finding the Greatest Common Factor At the beginning of this Lesson, when your teacher tells you, complete the common factor (sometimes called the GCF) is the highest factor shared by two or more numbers. That is, the biggest number that goes into any two numbers you could compare.

197 STUDENT MANUAL MATH LESSON 45-2 Step 1: List the factors of each number. Step 2: Find the common factors from each set. Step 3: Determine the highest number that is a common factor in the following problems: 1. Find the greatest common factor of 6 and Factors of 6 = 1, 2, 3, 6 Factors of 15 = 1, 3, 5, 15 Common factors = 1, 3 Since 3 is the highest common factor, this is the GCF (Greatest Common Factor). 2. Find the greatest common factor of 12 and Find the greatest common factor of 12 and 36. Complete the following Practice Problems to see if you understand how to determine the greatest common factor: 1. Find the GCF of 10 and Find the GCF of 18 and Find the GCF of 15 and 21 Lesson Wrap-Up: Create three pairs of numbers and trade these pairs of complete the Lesson 45 Practice Sheet.

198 STUDENT MANUAL MATH LESSON 45-3 Lesson 45 Practice Sheet Find the greatest common factor between each pair of numbers and and and and and and and and and and , 91, and and and and and , 54, and and and and , 90, and and and , 12, and and 4

199 STUDENT MANUAL MATH LESSON 46-1 Lesson 46 Finding the Least Common Multiple Factors and Multiple Assessment is, the smallest number that two numbers evenly divide into. First, review following sets: and and and and 49 The Least Common Multiple is the smallest of the common multiple of a set of numbers. Step 1: List all of the multiples of each number. Step 2: Find the smallest number that appears in each list. This is the LCM. Find the Least Common Multiple of 12 and 4. Find the Least Common Multiple of 6 and 8.

200 STUDENT MANUAL MATH LESSON 46-2 multiple of each and and and 7 Lesson Wrap-Up: Complete the Lesson 46 Practice Sheet.

201 STUDENT MANUAL MATH LESSON 46-3 Lesson 46 Practice Sheet Find the Least Common Multiple of each pair of numbers and and and and and and and and and and and , 24, and , 5, and and and and and , 7, and , 6, and and and and and , 9, and 15

202 STUDENT MANUAL MATH LESSON 46-4 Factors and Multiples Assessment Part 1: Complete each divisibility table. Write yes if the number is divisible by the given number. Write no if it is not divisible by the given number. 1. 6, , , by 3 by 5 by 4 by 3 by 4 by 5 by 7 by 6 by 4 by 8 by 10 by 5 by 6 by 9 by 11 by 6 by 10 by 10 by 13 by 11 Part 2: Classify each number as Prime or Composite Prime Prime Prime Prime Composite Composite Composite Composite Prime Prime Prime Prime Composite Composite Composite Composite

203 STUDENT MANUAL MATH LESSON 46-5 Part 3: Find the Greatest Common Factor of each set of numbers and and , 87, and and and and , 5, and and and 62 Part 4: List all of the factors of each number Part 5: Find the Least Common Multiple and and and , 10, and , 15, and and , 4, and and and 7

204 STUDENT MANUAL MATH LESSON 47-1 Lesson 47 Introduction to Exponents exponents, and being able to understand the meaning of a number written in exponential form is crucial. reading numbers in exponential notation: exponent and base. In the number 34, the 4 is the exponent and the 3 is the base. The exponent shows how many times the base is used as a factor in the problem, therefore, 34 means to multiply the base of 3 four times, or in other terms, 3 x 3 x 3 x 3 = 81., written in exponential form, and write the meaning of the number. Let s look at the following Examples: Example: 87 = 8 x 8 x 8 x 8 x 8 x 8 x 8 Example: 63 = 6 x 6 x 6 Example: The fourth power of two = 24 = 2 x 2 x 2 x 2 Example: 5 = 3 x 3 x 3 x 3 x 3 Look at the expanded form of a number, and write it in exponential form. We then will discuss the following Examples: Example: 4 x 4 x 4 = 43 Example: 5 x 5 x 5 x 5 x 5 x 5 x 5 = 57 Example: 3 x 3 = 32

205 STUDENT MANUAL MATH LESSON 47-2 Complete the following Practice Problems. Part 1: Write each number in exponential form: x 10 x 10 x 10 = 2. 7 x 7 = 3. (3y) x (3y) x (3y) = Part 2: Write the expanded form of each number: = = = Lesson Wrap-Up: Complete the Lesson 47 Practice Sheet.

206 STUDENT MANUAL MATH LESSON 47-3 Lesson 47 Practice Sheet Write the following in exponential form x 11 x 11 x 11 = 2. 3 x 3 x 3 = 3. 5 x 5 x 5 x 5 x 5 = x 10 x 10 x 10 = 5. 6 x 6 = 6. 2 =

207 STUDENT MANUAL MATH LESSON 47-4 Write using equal factors. Find each value = = = = = = Express in exponential form. 1. 1,000 = = = = = 6. 8 =

208 STUDENT MANUAL MATH LESSON 48-1 Lesson 48 Understanding the Expanded Form of a Number Let us begin by reviewing the concepts learned in Lesson 47. Complete the following Practice Problems: 1. Write 5³ in expanded form 2. Write the meaning of 6 x 6 x 6 x 6 3. Write 3² in expanded form 4. Write the meaning of 10 x 10 x 10 x 10 x 10 In this Lesson, you will use place value to write a number that is not in exponential form in its expanded form. In order to complete this, review the Place Value Chart explaining the number 5,641,203. Expanded Form Place Value 5 x 1,000,000 6 x 100,000 4 x 10,000 1 x 1,000 2 x x 10 3 x Millions Hundred Thousands Ten Thousands Thousands Hundreds Tens Ones

209 STUDENT MANUAL MATH LESSON 48-2 The expanded form of a number is simply the number multiplied by the count the number of zeros in the expanded form place value, and multiply the base by 10 to this power (the power representing the number of zeros). next page for your answers. The Milky Way Galaxy: miles across

210 STUDENT MANUAL MATH LESSON 48-3 This column should look like this: Exponential Form 5 x x x x x x x 100 Write each number below in expanded form using exponential Example: 1,569 = 1 x 1000, 5 x 100, 6 x 10, 9 x 1 = 1 x 10³ + 5 x 10² + 6 x x = Example: = 62,716 = Example: = Lesson Wrap-Up: Complete the Lesson 48 Practice Sheet.

211 STUDENT MANUAL MATH LESSON 48-4 Lesson 48 Practice Sheet Write each number in expanded form using exponents. 1. 3, , , ,239, , , ,898

212 STUDENT MANUAL MATH LESSON 49-1 Lesson 49 Prime Factors and Exponents Introduction to Exponents Assessment In a previous Lesson, you should have learned the difference between prime and composite numbers. If you are not sure which is which, ask your Practice determining whether a number is prime or composite with the following examples: number as a product of prime numbers.

213 STUDENT MANUAL MATH LESSON 49-2 Write a prime factorization by using the following Examples: Example: Write the prime factorization of 40. Step 1: Choose any two factors of 40. Step 2: Write these factors at the end of each line in the factorization tree. Step 3: Look at these two factors to determine if they are prime or each number. Step 4: Write these new factors at the end of each line in the factorization tree. Step 5: Continue this process until the bottom row of factors are prime numbers. Example: Write the prime factorization of 56.

214 STUDENT MANUAL MATH LESSON 49-3 Complete the following Practice Problems. 1. Find the prime factorization of Find the prime factorization of Find the prime factorization of 36 Lesson Wrap-Up: Complete the Lesson 49 Practice Sheet.

215 STUDENT MANUAL MATH LESSON 49-4 Lesson 49 Practice Sheet Find the prime factorization of each number

216 STUDENT MANUAL MATH LESSON 49-5 Introduction to Exponents Assessment Part 1: Express each number in its exponential or expanded form x 2 x x 3 x 3 x 3 x x 9 8. (10y) x (10y) x (10y) x 6 x 6 x x 8 x 8 x 8 x Part 2: Express each number in its expanded form using exponents , , ,120, ,287

217 STUDENT MANUAL MATH LESSON 49-6 Part 3: Create a prime factorization tree for each number and express it using exponents

218 STUDENT MANUAL MATH LESSON 50-1 Lesson 50 Finding Equivalent Fractions Suppose you were baking a batch of chocolate chip cookies and you needed to triple the recipe to feed a crowd of people. How would you determine the correct amount of each ingredient to put into the tripled recipe? With whole numbers, you would simply need to multiply by 3, but with fractions, finding equivalent fractions would be necessary. This is what we will work on learning in this Lesson. In order to create equivalent fractions, which are fractions that name the same amount, you must multiply or divide both the numerator and denominator by the same number. In the cookie Example, you would have to multiply the numerator and denominator of each fractional ingredient by 3 in order to triple the value of the ingredients. You could also use a number line to determine equivalent fractions. In the following Examples fill in the missing number: Example: 3 / 4 =? / 16 Example: 9 / 27 =? / 3 Example: 5 / 15 = 15 /? In the first example you should be able to see that the 4 is being multiplied by 4 to create 16 in the second fraction. Therefore, to find the missing number, you should multiply 3 by 4 to get 12 as the equivalent fraction. In the second example you should be able to see that 27 is being divided by 9 to create 3 in the second fraction. Therefore, to find the missing number, you should divide 9 by 9 to create a new fraction of 1/3. In the third example you should see that the 5 is being multiplied by 3 to equal 15 in the second fraction. Therefore, multiplying 15 by 3 will give a missing value of 45, creating an equivalent fraction of 15/45.

219 STUDENT MANUAL MATH LESSON 50-2 You can place equivalent fractions on a number line. Equivalent fractions are on the same place on a number line. Here are the examples on a number line: 0 Complete the following Practice Problems. Make sure you follow the directions carefully. Create an equivalent fraction by following the given operation / 8 multiply by 2 1/3 9/27 5/15 15/ / 12 divide by / 12 multiply by 3 /36 Place each of the above fractions and the equivalent fraction on the number line

220 STUDENT MANUAL MATH LESSON 50-3 Fill in the missing number in each equivalent fraction set / 5 =? / / 8 = 9 /? 3. 7 / 12 =? / 24 Place each of the above fractions and the equivalent fraction on the number line Lesson Wrap-Up: Complete the Lesson 50 Practice Sheet.

221 STUDENT MANUAL MATH LESSON 50-4 Lesson 50 Practice Sheet Fill in the missing number to create a pair of equivalent fractions. 1. n 2. n 3. n 4. n n = n = n = n = 5. n 6. n 7. n 8. n n = n = n = n = 9. n 10. n 11. n 12. n n = n = n = n = 13. n n 16. n = n = n = n = n n n = n = n = n =

222 STUDENT MANUAL MATH LESSON 50-5 Choose 2 sets of equivalent fractions and place them on the number line PRINT

223 STUDENT MANUAL MATH LESSON 51-1 Lesson 51 Reducing actions Fr, below: and and and 60 In this Lesson, you will learn how to write a fraction in its simplest form, which is the form of a fraction in which the numerator and denominator have 1 as the only common factor. Putting a fraction into its simplest form is often called reducing. In order to put a fraction in its most simple form, you follow these steps: Step 1: Find the greatest common factor between the numerator and denominator. Step 2: Divide the numerator and denominator by the greatest common factor. Step 3: The new fraction will be in the simplest form.

224 STUDENT MANUAL MATH LESSON 51-2 Complete the following Examples to explain how to reduce fractions: Example: 4 8 = ½ Example: = 3 5 Example: = 5 8 Complete the following Practice Problems. Determine the greatest common factor of the numerator and denominator, and use this number to reduce the fraction to its simplest form: Lesson Wrap-Up: Complete the Lesson 51 Practice Sheet.

225 STUDENT MANUAL MATH LESSON 51-3 Lesson 51 Practice Sheet Reduce each fraction to its simplest form

226 STUDENT MANUAL MATH LESSON

227 STUDENT MANUAL MATH LESSON 52-1 Lesson 52 Mixed umbers N Review the following word problem: Sammy went to the donut shop to buy donuts for his classmates. He has 27 students in his class, including himself, so he wants to buy 27 donuts. Since the donuts are sold by the dozen, Sammy will need 2 dozen donuts, and 3 extras (2 dozen is 24 donuts). This amount of donuts can be represented using the mixed fraction Mixed fraction A fraction consisting of a whole number and a fraction. Mixed fractions represent fractions greater than 1. Improper fraction A fraction where the numerator is larger than the denominator. fraction into an improper fraction. To do this, follow these steps: Step 1: Multiply the whole number by the denominator. Step 2: Add this product to the numerator. Step 3: This sum is the numerator of the new fraction. Step 4: The denominator of the new fraction stays the same. These are Examples of converting a mixed number to an improper fraction: Example: Step 1: Multiply the whole number of 3 by the denominator of 5 to get 15. Step 2: Add the product of 15 to the numerator of 4 to get 19. Step 3: Put 19 in the numerator and 5 in the denominator to create the improper fraction 19 5.

228 STUDENT MANUAL MATH LESSON 52-2 Example: Step 1: Multiply the whole number of 1 by the denominator of 8 to get 8. Step 2: Add this product of 8 to the numerator of 7 to get a sum of 15. Step 3: Put 15 in the numerator and 8 in the denominator to create the improper fraction Example: 10 ½ Step 1: Multiply the whole number of 10 by the denominator of 2 to get a product of 20. Step 2: Add this product to the numerator of 1 to get a sum of 21. Step 3: Put the 21 in the numerator and the 2 in the denominator to create the improper fraction 2½. Complete the following Practice Problems by converting each mixed fraction to an improper fraction: Now, you will also learn how to convert an improper fraction into a mixed number. A solid understanding of division is necessary to complete these types of problems. You will have to follow these steps: Step 1: Divide the numerator by the denominator. This number indicating how many times the numerator can be divided into the denominator is the whole number in the mixed number. Step 2: Determine the remainder of the division problem. This number will be the numerator of the mixed number. Step 3: The denominator of the mixed number is the same as the denominator in the improper fraction.

229 STUDENT MANUAL MATH LESSON 52-3 Work through the following Examples on how to convert an improper fraction to a mixed number: Example: 15 4 Step 1: Divide 15 by 4 to get a whole number of 3. Step 2: The remainder of this division problem is 3. This is the numerator of the fraction. Step 3: The denominator of the fraction is 4 (the original denominator). Therefore, the mixed number is 3 ¾. Example: 19 2 Step 1: Divide 19 by 2 to get a whole number of 9. Step 2: The remainder of this division problem is 1. This is the numerator of the fraction. Step 3: The denominator of the fraction is 2 (the original denominator). Therefore, the mixed number is 9 ½. Example: 34 5 Step 1: Divide 34 by 5 to get a whole number of 6. Step 2: The remainder of this division problem is 4. This is the numerator of the fraction. Step 3: The denominator of the fraction is 5 (the original denominator). Therefore, the mixed number is

230 STUDENT MANUAL MATH LESSON 52-4 Complete the following Practice Problems to practice converting improper fractions to mixed fractions Lesson Wrap-Up: Complete the Lesson 52 Practice Sheet.

231 STUDENT MANUAL MATH LESSON 52-5 Lesson 52 Practice Sheet Part 1: Convert each mixed fraction to an improper fraction ½

232 STUDENT MANUAL MATH LESSON 52-6 Part 2. Convert each improper fraction to a mixed fraction

233 STUDENT MANUAL MATH LESSON 53-1 Lesson 53 Comparing and Ordering Fractions Can you answer the following question? Amanda ordered a small pizza and ate 3 6 of it for dinner last night while her sister, Anna, also ordered a pizza and ate 2 3 of her pizza for dinner. Which sister ate the most pizza for dinner? In the above Example, understanding how to compare two fractions with unlike denominators will help in answering the question. In order to compare fractions, it is often helpful to convert the fractions to common denominators and then compare the numerators to see which fraction is larger A common denominator between 6 and 3 is 6, therefore, we will convert each fraction to sixths, but since 3/6 is already in sixths, we only need to convert x 2 = 4 6 Making this conversion, we can see that 4 6 is larger than 3 6, so Anna ate more pizza than Amanda. You can use the same process to compare and order mixed fractions as immediately help you answer the question.

234 STUDENT MANUAL MATH LESSON 53-2 Let s discuss the following Examples: Compare each fraction using < or > Convert each fraction to common denominators. A common denominator is will become and 4 5 will become Therefore, looking at these fractions you can see that 5 7 < Convert each fraction to common denominators. A common denominator is will become 4 12 and 2 4 will become Therefore, looking at these fractions, you can see that 1 3 < ¾ You do not need to make any conversions in this problem because if you examine the whole numbers in each fraction, 3 is larger than 2, therefore, > 2 ¾. Order each set of fractions from least to greatest , 2 3, ½ Convert all three fractions to common denominators. The common denominator is 30. Therefore 4 5 becomes 24 30, 2 3 becomes 20 30, and ½ becomes Looking at these three fractions, you can see that the order from least to greatest should be ½, 2 3, 4 5 (15 30, 20 30, 24 30) ½, 4 5 7, 3 ¾ Convert all three fractions to common denominators. The common denominator is 28. Therefore, 3 ½ becomes , becomes , and 3 ¾ becomes 3 2½8. Looking at these three fractions, you can see that the order from least to greatest should be 3 ½, 3 ¾, and , 18 9, 3 9 Convert all three fractions to common denominators. The common denominator is 63. Therefore, 6 7 becomes 54 63, 18 9 becomes , and 3 9 becomes Looking at these three fractions, you can see that the order from least to greatest should be 3 9, 6 7, and Lesson Wrap-Up: Complete the Lesson 53 Practice Sheet.

235 STUDENT MANUAL MATH LESSON 53-3 Lesson 53 Practice Sheet Part 1: Compare each pair of fractions using < or > ½ 3. ¼ ¼ ¾ ½

236 STUDENT MANUAL MATH LESSON 53-4 Part 2: Put each set of fractions in order from least to greatest ¼ ½

237 STUDENT MANUAL MATH LESSON 54-1 Lesson 54 Fractions and Decimals In this Lesson, you will practice converting fractions into decimals. You ve probably done this before, but it will be good for you to practice since later Lessons will need you to make these conversions more and more. To convert a fraction to a decimal, you divide the denominator into the numerator. In other words, divide the number on the top by the number on the bottom. Unless the fraction is mixed or improper, you ll get a value of less than one. This means you will have to place a decimal. Since a decimal always goes to the right of the ones place, you ll find that most of the time your answer begins with a decimal point. Here are several Examples of converting a fraction into a decimal. Go through these with your teacher, and then complete the Lesson 54 Practice Sheet: Example: 4 / 10 Example: 3 / 5 What happens if you convert two equivalent fractions into decimals? Convert the following equivalent fractions into decimals: ¾ 3 / 4 =.75 and 6/8 6 / 8 =.75 Equivalent fractions convert to equal decimals. Lesson Wrap-Up: Complete the Lesson 54 Practice Sheet.

238 STUDENT MANUAL MATH LESSON 54-2 Lesson 54 Practice Sheet Write each fraction as a decimal. Round off to the nearest thousandth if needed

239 STUDENT MANUAL MATH LESSON 54-3 Write each set of equivalent fractions as decimals /5 and 12/ /10 and 10/ /8 and 4/ /8 and 14/ /9 and 2/ /7 and 15/21 PRINT

240 STUDENT MANUAL MATH LESSON 55-1 Lesson 55 Fractions and Decimals Assessment In Lesson 54, you learned to convert fractions into decimals. Now, you will convert decimals into fractions with denominators of 100, and then reduce them. The conversion of a decimal to a fraction is as follows: Step 1: Determine the place value of the last digit in the decimal. Step 2: Use this place value as the denominator for the fraction. For Example, if the place value is hundredths, the denominator will be 100. Step 3: Use the number without the decimal point as the numerator and drop any leading zeros. Work through these Examples of converting a decimal to a fraction: Example: 0.05 Example:.457 In Lesson 54, you had to round off some of your decimals to the nearest thousandth. They may have been too big, or they may have just gone on forever. Some decimals end on a neat, even number these are called Terminating Decimals. Those that go on forever are called Repeating Decimals.

241 STUDENT MANUAL MATH LESSON 55-2 Terminating decimal A decimal that would give no remainder if you were to use long division..80 is an Example of a terminating decimal. Repeating decimal A decimal which does not end, and makes a pattern of numbers over different place values is an example of a repeating decimal because the 36 continues to repeat. To represent a repeating decimal, place a bar above the numbers that repeat. You would write this example as:.36 Convert these decimals to fractions: = = = Lesson Wrap-Up: Complete the Lesson 55 Practice Sheet.

242 STUDENT MANUAL MATH LESSON 55-3 Lesson 55 Practice Sheet Part 1: Write each decimal as a fraction Part 2: Write each fraction as a decimal. Indicate if it is a terminating decimal with a T or a repeating decimal with an R ¼ ½ ¾ 21. ¾0

243 STUDENT MANUAL MATH LESSON 55-4 Fractions Assessment Part 1: Order the fractions from least to greatest ½ Part 2: Write each fraction in simplest form

244 STUDENT MANUAL MATH LESSON 55-5 Part 3: Write each improper fraction as a mixed number in simplest form Part 4: Write each mixed number as an improper fraction ½

245 STUDENT MANUAL MATH LESSON 55-6 Part 5: Convert each fraction to a decimal ninety-seven and eight tenths hundredths forty-one hundredths tenths Part 6: Convert each decimal to a fraction

246 STUDENT MANUAL MATH LESSON 56-1 Lesson 56 Addition and Subtraction of Fractions with Like Denominators Oftentimes, in real life, the ability to add and subtract fractions is crucial. Examine this Example: Mary ate 1/10 of the pizza her family ordered for dinner and her brother ate 5 /10 of the pizza. How much of the pizza did both eat together? SA When adding and subtracting fractions, it is important to understand important terminology, such as the words numerator and denominator. The numerator of a fraction is the top portion of a fraction, and the denominator of a fraction is the bottom portion of a fraction. M If the two fractions being added or subtracted have the same denominator (or bottom portion), the steps for completing the problem are simple: PL Step 1: Add or subtract the numerators. Step 2: Bring the denominator down into the answer. E